Doomsday happens to be one of those champions I seem to have a lot of fun with. Yes, the Doomsday, Superman's clone gone wrong. As a mean monstrosity on the field, Doomsday's skill set is brutally straightforward, for want of a better word. Pummelling Assault simply means Doomsday throwing a tantrum around the enemy, jumping on the ground three times. Berserk gives him a boost in attack speed and movement speed, while damaging enemies in the vicinity for 4 seconds. Charge is another one of Doomsday's melee skills, dealing damage by carrying an enemy off. Lastly, Doomsday's ultimate skill, or ulti for short, is Kryptonian Rage, which gives additional attack damage, attack reflect and resilience until his Fury runs out, which can be regained through either normal attacks or skill use.
Being someone who can only take the fight up close and personal, I usually emphasised on Doomsday's health and attack such that he can take as much as he can dish out. Equipment for endurance in the fray include Suit of Sorrows, which grants 30% attack reflect, complementing well with Doomsday's Kryptonian Rage; Kryptonian War Armor, for additional health regeneration and move speed; and Ruby of Life, again for additional health.
For attack, I would go for Lobo's Chain, which adds on both attack and health, with an added effect of slowing down an enemy with basic attacks; either Claw of Horus or Soultaker Katana, both boosting attack as well as granting a lifesteal effect for basic attacks; and finally, Deathstroke's Claymore (because hey, Deathstroke), if I manage to rake up enough money for it, which gives additional attack and attack speed. There are still other items I haven't tried out yet, though considering the number of times I've sent Doomsday charging into the chaos of battle equipped as such and managing to come out almost unscathed, the changes I would make in this equip would probably be scarce, at least, assuming there won't be any drastic alterations to the item store in the near future, where the game finally comes out of beta mode.
As Doomsday, I would often avoid from long-range attackers, such as Green Arrow and Mecha Wonder Woman, early in the game at least. As two of Doomsday's offensive skills happen to have an area effect, he would go wherever the throng is, as in, wherever most of the players are fighting it out. Since he doesn't rely at all on stamina or mana or what have you, only thing standing in the way of spamming skills at others is his cooldown rate and second-long delay between skill use. What's more is that you can use Doomsday's ulti almost frequently, with the condition of gathering up full Fury, which can be done easily given enough drones around to trample on.
I would say that Doomsday is a relatively easy champion to play; as long as you can balance between health and attack, he can be turned into a force to be reckoned with, as he deserves no less.