The Incredible Hulk

There are 12 different classes of attacks for this achievement.
You can view them and how many kills you have with each type by going to the Pause Menu > Stats > Hulk - Victories.

Unarmed - Impossible to miss, any basic attack using your fists.

Thunderclap - While unarmed, press X then Y for a basic thunderclap. It's a bit tricky, since you'll just kill basic enemies with the first hit, so try it on tougher enemies or try to miss the first hit on purpose. I assume the Thunderclap Rage attack would also count, but you have to unlock it.

Ground Slam - X, X, Y. Try this move on the small mech enemies and you'll get a kill soon enough. I assume the Ground Slam Rage attack would also count, but you have to unlock it.

Shield - The game teaches you how to use a shield in one of the early missions. You can hold X + A to dig a shield out of the pavement, then just smack someone around with it.

Sprinting - You have to unlock sprinting, you can see the requirements for this in the Pause Menu > Upgrades. Once you have it, it's easy to simply sprint into a weak enemy and kill him.

Car - Pick up a car and smash it at some enemies. The Y attack is very effective against heavier enemies.

Pole - Find a street light or something similar and press B to grab it, then use it on a weaker enemy for an easy kill.

Mace - The easiest way to get a mace weapon is to find a thick tree (try Central Park) and smash it, search the pieces for one with a big clubby end to it. Pick it up and you'll use it as a mace, find an enemy and smash him.

Rage - You need to unlock these, there's the "Improved Thunderclap" and the "Improved Ground Slam". You can see the unlock requirements in the Pause Menu > Upgrades. Once you have one, get enough rage by fighting/destroying and unleash one of them on some enemies.

Aerial - There are a couple of different aerial attacks, jumping then using X or Y are probably the easiest. But you can do aerial attacks while holding poles, cars, maces, etc. as well.

Grab - Pick up a small enemy and wail on him with X.

Fall damage - I'm not 100% sure, but this seems to mean simply falling on an enemy to kill them without using an actual attack. Try to get some weak enemies following you, go up a building and fall on some of them.