You can look at the Awards showcase and see which ones you have unlocked. From the Main Menu or Pause screen go to Barracks > Outbreak Card. There are a total of 37 awards to be unlocked. The awards are varied and include versus multiplayer awards, kill streak awards, survival awards, level completion awards, and multi kill awards.
By the time you finish a playthrough of Story mode, you will have unlocked around 15 awards which include level completions, kills streaks and most if not all of the weapon kill streak awards. For the weapon kill streak awards, getting 10 kills without going down on most weapons isn't too hard, as long as you stick to cover and aim for headshots to save ammo and increase accuracy. You may have trouble with getting the 10 melee, grenade, and shockwave skills therefore these can be boosted with another player in a private Deathmatch online.
Now you must work on your multi-kills, this includes a quadruple kill with one Grenade, adhesive grenade, and shockwave ability. These can be quite tricky to pull off in the story mode and there aren't any known special places to get these. However, these can also be boosted in multiplayer if you get 5 people together. Just start a private Deathmatch and have everyone group up together for whoever is going for the kill.
Now you should work on the multiplayer versus specific wards, there are two related to Capture the Flag, 2 for Team Deathmatch and another 2 where you must end a killing spree and kill someone after you have died. Again, these can all be boosted in private matches with one other player.
Now you have the killing blow awards left. For these you must be the one to perform the final blow to all 3 major bosses in the game. You fight the Digger in level 3 section 6, then Lilith on level 4 section 6, and the Banshee on level 4 section 7.
Finally all that remains are the survival awards which require you to complete each level without getting incapacitated. Easily the hardest bunch of the awards. Using the final section exploit does not unlock the award here so your best bet is to do this in 4 player co-op, 2 people do most of the fighting while the other 2 that are going for the awards try to stay back and alive. You will probably want to play on Casual and be careful when it comes to the bosses.