Time for a guide to who I think is one of the most fun solo laners in the game. Doomsday has a great lane clearing ability and you can land lots of fast skill damage without using a basic attack to draw minnion aggro. He is also the ideal candidate for zoning the enemy out. With fast regen, and all the damage he can give and receive, he'll keep that wave almostly constantly under the enemy tower.

Let's first talk about Stolen Powers. I personally like taking Robin's Cameras, and Catwoman's Teleport. The first is so that you can have vision of any possible ganks, and you'll be in lane a very long time if you are playing Doomsday right so you'll need extra charges. I changed to taking teleport again just for the presence across you can have across the map. 

Let's take a break, and talk about amps. You want Empowered Suit of Sorrows and Mirror Master's Prism. Your strength come from being in the center of the enemy team and just being a menace. These mods allow you to do that extremely well. From there you have Hook Shot on Lobo's Chain and +10 AD/Term 2 on Claymore to consider. Use the mod that best fits what you want to do with him. I would only take the +AD if you were building it as your second main item. 

Now onto your items. Pick the Nil item you want at the start. I have them both up there. Choose one and 3 health pots at the start. You want to build up Lobo's chain as fast you can to T3. From there you can go the damage route or build into your reflect item(s). If you were going the damage route and modded +10 AD on claymore, pick it up right now and leave it at T2. Even if you bought Term 2 on Claymore, I would build into the reflect tiem to counter the main damage I was recieving before capping Claymore. 

Once you have all your core items built. it is time to choose optional cap items based on what fits best. I like Atomic Axe for its massive penetration. Doomsday naturally has a very high base damage ratio.  Using the Atomic Axe will make that high base work wonders for you, but almost any combination of those items would be a great kicker in a Doomsday build.

The How: You want to max W and E fast. Q slows down a chase and doesn't do a lot of damage anymore. It is great using after you push someone into a wall though. W gives you bonus AD and is like an extra item at cap. A W into an E into a wall followed by three successful Q does a lot fo damage to anyone. It works well for trading in lane when your enemy is out of position and for ganking your other lanes. Just remember to never push someone away from your team or back under their turret. It is just bad. 

Remember to roam gank (after level 6 mostly) and steal enemy jungle every time you are in position to. This will not only net you more farm/credits, but also slowly put the enemy team behind just from denial.