Starter items

Core items

Offensive items

Defensive items

Situational items

Stolen Powers
Jungle
Nightmare Batman's Consume provides Batman with a good degree of sustain, the earlier high heal more beneficial than Mecha Superman's, as much of Batman's HP items are built later. Late-game it remains useful because of Vitality from Kryptonian War Armor.
Thematically, Nightmare Batman is Batman from the Nightmare universe, so, consume away.

Joker's Detonate gives quick wave clear and can be used against champions as a slow and extra burst of damage.
Thematically, Joker is Batman's nemesis, and he's stolen stuff from Batman, so return the favor.

Lane
Green Arrow's Surveillance Camera lets you destroy enemy cameras in stealth pads, allowing you to reliably make use of them for your passive's bonus damage.
Thematically, this ties in to billionaire crime-fighting geniuses like Bruce Wayne and Oliver Queen sharing technology.

Superman's Super Strength provides you with an extra ranged attack against your enemies while also providing you with increased attack damage, which can help offset being pushed out of stealth pads, preventing you from taking advantage of your passive.
Thematically, it fits with Superman being one of Batman's closest friends and primary team-up partners as well as tying into the Kryptonian War Armor you will build.

When laning, switch the leveling order of your W and E.

Augments
Platinum, because of the Platinum Arkham City skin, works with Batman's Utility Belt and Deathstroke's Claymore to drop your cool-downs a little further, allowing Batman's ultimate to be used more than once every 70-seconds.

Alpha and Omega, because Batman is the Beginning and the End, provide him with some extra health early and late game, to help keep him alive through early ganks and put him at just over 2900 HP late-game at max level and full build.
The penetration of Alpha and Omega combine with Platinum to start you with 5.4 attack penetration, allowing for some early bite.

Diver, because you'll be diving people in their towers and murdering them and walking out like a boss, gives you some extra late-game damage to take advantage of the buff from your passive. No matter where your enemy tries to hide, you will look for them, you will find them, and you will kill them.
Diver's bit of extra attack life-steal provides Batman a little more sustain, either in jungle or lane, and helps make up for taking the lower attack life-steal of Atlantis Gambit vs that of Cheetah's Claw or The Soultaker.

Artifacts
Starter
In lane, take Nil Weapon, Health Vial 2 and Atlantis Gambit 1.
In jungle, take Maurader Knife 2 and Atlantis Gambit 1

Atlantis Gambit increases Batman's attack damage and gives some needed HP regeneration in addition to attack life-steal. Because of Batman's higher HP and armor with this build, he can remain in team-fights for extended periods, leading the team and providing its beneficial aura for the duration of the fight. The Targeting System keyword at Tier 4 allows for easy ganks, as its long range can let you set up for allies before you even reach the fight or slow an enemy from a distance so you can close in. The additional healing on basic attacks makes great use of Batman's high HP from this build, especially late-game. Further, the Tier 2 mod adds very cheap and early armor to help with damage trades.
In comparison, Cheetah's Claw gives a little more attack life-steal, but without the aura, and raises Batman's attack speed higher than 1 per second, disrupting timed use of skills to add procs from Batman's Utility Belt to your basic attacks.
The Soultaker does provide higher damage and life-steal for Batman, but sacrifices the attack life-steal aura and Targeting System keyword.
Thematically, this artifact fits with Batman being a man of the Earth, Superman being a man of the skies and Aquaman being a man of the seas, and Bruce is already going to be getting his Utility Belt and some Kryptonian gear, while Targeting System makes it feel like you're spying on enemies with satellites or something., your reactions slowing their movements to a crawl as you prepare to attack. Also early armor as Batman prepares himself in anticipation of enemy armor penetration.

Deathstroke's Claymore provides high attack, penetration and cool-down, in addition to its Terminate keyword, which allows you to carve through tankier targets or still have an advantage even when not in stealth or using your ultimate.
In comparison to other similar cost items, Coda Blade Tier 3 costs 1750 credits and provides 40 attack and 10 penetration, where as the modded Deathstroke's Claymore Tier 2 costs 1750 credits and provides 35 attack, 10 penetration and 10% cool-down reduction.
Thematically, this weapon ties in to not only Batman's training amongst the League of Assassins and his other martial training but also into swordsmanship being Batman's most honed weapon skill, in addition to Deathstroke being a recurring enemy of Batman's, while Terminate ties into Batman's ability to strike precision blows. And it's a bad ass sword, so put more damage on it.

Batman's Utility Belt provides him with HP, attack damage, cooldown reduction and bursts of damage from its skill effect, a helpful bonus as Batman's base damage reaches 100 at max level. The move speed mod adds the extra 10 that separates Batman from the Flash's base, and puts him a bit faster than Superman and most other Bruisers, allowing for those narrow escapes or catches.
Thematically, it's Batman's Utility Belt. Give it to him. And while a core item, no mod, because it's Batman's Utility Belt. It's bad ass enough as it is.

Claw of Horus provides a boost of HP, attack armor and damage, overall improving Batman's capabilities as a melee champion. The armor strip allows him to deal with even thicker armored targets, gradually stripping their defenses away while building up his own, allowing him to last in the fight even longer. The mod allows for a total of 45.4 attack armor penetration if you fully strip a target, in addition to providing you with 25 more attack armor, countering other champions' penetration items.
Thematically, this fits in with Batman learning his opponent's strengths and weaknesses as he fights, improving his defense against their attack while landing more critical blows of his own. Also, Batman's really good at analyzing his enemies, so enhance the armor steal.

Lobo's Chain increases Batman's HP and attack damage, while the slow better enables him to stick to a single target in order to assassinate them, especially if he catches someone in a stealth field and can slow them enough to land a few more damage boosted hits. Hook-Shot provides a single-target counter-part to the group-wide cripple of his ultimate. He can cripple the enemy team and jump down on a foe, crippling them again before they can get away.
Thematically, Batman uses grappling hooks, so why not an awesome hook-chain instead that lets him slow his targets with winding blows to the body?

Kryptonian War Armor further boosts his HP and brings his power armor up, providing some much needed protection against inevitable power skill spam used to flush you out of stealth pads, in addition to simple late-game power damage countering. The HP regen ties with Atlantis Gambit's to give a great boost to Batman's sustain, especially with its Vitality keyword boosting the effect of Atlantis Gambit's attack life-steal aura. The extra move speed enhances your ability to avoid skill-shots as well as your ability to catch a fleeing target or make your own escape, while also cutting down your travel time around the map.
Thematically, it fits with Batman needing awesome armor if he hopes to fist fight someone like Doomsday or Superman. Also, he and Superman are close, so Kal won't mind Bruce using his stuff.

Situational
Metallo's Heart can be a good replacement for Lobo's Chain against damage-centric teams, as while you give up some damage and the slow, you gain increased armor against their damage and the aura combines with Batman's Cape Sweep to provide a 4-second 50% attack speed debuff to all affected enemies, allowing you to outright prevent a large amount of damage from being done. This can also allow you to more precisely time your skills so you can land a basic attack with Batman's Utility Belt proc between each one, increasing your overall damage. The additional cooldown of Metallo's Heart combines with the rest of the build to allow an ultimate more than once per minute.
Thematically, as a piece of Kryptonite, this can be extremely helpful against a rampaging Supergirl, or any other hyper steroid attack speed built attack champion. However, as a piece of Kryptonite, it is more of a last resort weapon of Batman's, thus its situational use.

Atomic Axe can replace Lobo's Chain against armor-centric teams, as heavy armor builds tend to have low damage, so the exchange in HP is made up for in sheer armor penetration and damage. A total of 45.4+35% penetration.
Thematically, Batman is an expert with many forms of weaponry. However, as Batman prefers unarmed combat and makes extensive use of grappling hooks(Lobo's Chain), in addition to his preferred/best weapon being a sword(Deathstroke's Claymore), this is a situational item.

Build Order
1.
Atlantis Gambit 2. 500 credits.
2.
If laning solo, Deathstroke's Claymore 2. 750 + 1000 credits.

If laning duo, Deathstroke's Claymore 1 then Atlantis Gambit 3. 1750 credits. 750 + 1000 credits.

If jungling, Atlantis Gambit 4. 1000 + 750 credits.
3.
If laning solo, Atlantis Gambit 4. 1000 + 750 credits.

If laning duo, Deathstroke's Claymore 2 and Atlantis Gambit 4. 1000 + 750 credits.

If jungling, Deathstroke's Claymore 2. 750 + 1000 credits.
4.
Claw of Horus 3. 750 + 1000 + 1250 credits.
5.
Deathstroke's Claymore 3. 1500 credits.
6.
Lobo's Chain 4. 500 + 750 + 750 + 1000 credits.
7.
Kryptonian War Armor 3. 750 + 750 + 1250 credits.

Combat
Initiate from Stealth with Martial Arts, activating the proc on Batman's Utility Belt, and Batman lands a basic attack afterwards applying the Belt damage, stealing armor and slowing them. Follow with Cape Sweep to proc your Belt and land another basic attack. Target them with Batarang and basic attack, providing you with enough energy for your second Martial Arts, so whether or not they try to escape you can follow and basic attack for the kill, or finish with your ultimate if necessary.

After killing an enemy with your full combo your first Martial Arts charge will have finished cool-down, your second will be restored by the kill, and both Cape Sweep and Batarang should be off or finishing cool-down, with your Energy regenerating quickly to full and your damage boosted due to your ultimate, allowing you to engage another enemy with Atlantis Gambit's Targeting System to help you heal back up with your ult-damage boosted combo or to make a quick escape.

If the enemy uses an escape move after you first initiate with Martial Arts and your basic attack, wait a moment before using your second Martial Arts charge, landing another Bat's Belt procced basic attack. Follow with Cape Sweep to slow their counter-attack and land a Bat's basic attack before hitting them with Batarang and Bat's basic attack for the kill, or finish with your ultimate if necessary.

If at any point in time during your combo you notice the enemy preparing to use their ultimate against you, use your's to escape it and jump down on them, finishing them off with whatever skills were left in your combo chain.

Because of keeping your basic attack speed under 1 per second, not only can you make efficient use of the proc from Batman's Utility Belt but also the extra move speed from Kryptonian War Armor to help you avoid enemy skill shots in the small delay between landing your own moves.

Jungle Clear
Start with a Cape Sweep to the group as you walk towards the leader and work him with basic attacks if its the quad or trio monster camps, or Batarangs and basic attacks if its the mid camp.
The double monster camps can be finished more quickly early on with a single Martial Arts against the minion, while you basic attack and Cape Sweep the leader. The travel time to and from these camps gives your Martial Arts time to cool-down.
Once your damage and penetration is up some, you should be able to finish any non-mid camp with your second Cape Sweep.

Summary
This build provides that feeling of Batman being able to lead his team and jump into the middle of a group of enemies and defeat them one after another, while surviving any hits they throw his way either because of his agility(avoiding skill-shots) or the quality of his armor and ability to roll with the punches(armor/HP), in addition to being able to catch a second wind(attack life-steal/Atlantis Gambit's Targeting System). Or while stalking on his lonesome, be able to jump out at a vulnerable enemy and tear them apart in a quick flurry of blows, before he's off in the shadows again. "Where are you!?" Here. "Aaah!"

Also, no Vel9 Implant because of the problem of raising his attack speed over 1 per second, disrupting the use of procs from Batman's Utility Belt, and no Joe Chill's Revolver because Batman would never use a gun, least of all the one that killed his parents.