To check your status on how many you defeated, hit and then for statistics. It will show how many you have done. You can replay levels and challenges. This must be done on one character and cannot be split in different save files.
The following below will describe the various enemies you need to kill to get the "defeat X amount" achievements.
Purifiers (Challenge 9): These are the more common enemy in the game. If you need more, you can replay challenge 9. There are 100 enemies as well as a Stalker.
Stalkers (Challenge 8 or 11): This enemy is a mech that stomps and shoots guns. You must attack it's legs, then attack it when it falls. The best challenge for this is challenge 8 or 11. Challenge 8 has a U-Man as well, so it is the better choice.
Enforcers (Challenge 9): This enemy is a mech with a force field around it. You must destroy the force field before it's health can be depleted. The only challenge these appear on is 9.
U-Men (Challenge 7 or 8): This enemy can be normal or giant. There are a good amount to kill on challenge 7, but you can get some while you go for stalker kills on challenge 8.
MRD (Challenge 15): This is an enemy you see more during the final stages in the game. Play challenge 15 to get these kills. An alternate method is when you are fighting a lot of MRD with The Brotherhood, do not continue on. The enemy will keep respawning and you will be able to use your ultra power often to rack up fast kills.
Note: Sometimes you may have to wait a long time before your achievement will unlock for enemies defeated. Example would be defeating 2388 purists before the achievement finally unlocks. Another is defeating 22 enforcers, then the achievement unlocks 15 minutes later while you are fighting normal enemies.