Aquaman is insane. His max trident range damage comes from 100% of his power damage, which with a short follow up will make quick work of any squishy champions. 

First let's go over his skills.

Passive- Rising tide
After Aqua uses a skill he gains rising tide, which deals power damage and gives him health on his next basic attack.

Q- Trident of the king
Your bread and butter skill. Skill shot that deals power damage based on distance thrown. Has a long range and marks the target wit telepathy, which reveals them.

W- Stagger
Deals power damage and knocks the enemy down for one second. Note: cooldown is not shortened when upgraded.

E-  Undertow
Aqua dives forward a bit, dealing attack damage. Only ability that deals attack damage. Can be used again after hitting an enemy. Also good for clearing drone waves, chasing, escaping, and for two instant rising tides.

R/Ult- Might of Atlantis
Locked until damage is dealt with a skill, and is different depending on the skill and has a low 30 sec cooldown
Q- Surge. Aqua jumps to the target marked with telepathy by his trident, dealing power damage.
W- Wrath. Aqua dives on top of an enemy, dealing power damage and a percentage of the target's missing Hp. Can be targeted someone besides who was staggered. Good finisher and with low cooldown on ult you should be using it in most 1v1s.
E-Capsize. A third undertow that flips the enemy behind Aqua. Good for ganks or stopping an enemy from retreating to a control point but not much else.

Starting by grabbing two face's 1, doc destiny's 1, a radion shard (1), and the logoz 1. Aqua is fine bot or top but I usually go bot to start stacking my logoz early.  Rush the second tier of the logoz then the coin for free power damage. From there you can get how much you can afford, but save the scarab until you've got the starheart. When fighting in 1v1s start with the trident of course, then walk back to be sure they can't counterattack. Then if you're feeling confident ult them and follow up accordingly. Aquaman can be kited pretty easily, so if you're chasing someone on low health simply walk away to gain range on your trident and finish them off. If the trident isn't enough and it's off cooldown ult them. For another combo you can also just walk at them after the trident (Watch out doing this against a fed marksman, since they can start attacking you before you can attack them), E them (remember to attack between E's to use your rising tide), W them depending on their health (don't do it if they're on high hp) then ult to finish them. My mods for aqua are the 2 stacks of damage for the logoz collector, the extra power damage % used in the scarab, and the extra power damage on the book. The amps can just be various power damage ones with some cdr or lifesteal.