Rather than completing the game three times, simply play it under the Insane difficulty to get the following achievements at once: all three achievements for each act, completing the game, the Broken Fingers, and A Dish Best Served Cold.
When playing online, search for a one round Warzone game with a round length of 1 minute. This is called an achievement match and can get free kills towards your achievements. Each player takes a turn. One person from each team will go to the opposite side of the map while his teammates stand behind and wait for a person from the other team to come over to them. The one player from the other team kills everyone, except for the one person who is killing on the other team. The other players must stand there and let him kill them any way desired. If you swing your smoke grenade or throw it, the person will down your teammates and you can revive them, so you can get your revive achievement. When there is only one person on each team remaining, wait for the time to run out so you can start again, with another person on your team turn taking their turn to unlock the achievements.
When starting a Co-Op campaign, instead of logging into your main Gamertag as player one, log in as player two. Doing this from the beginning or while replaying chapters will let you play as Dominic Santiago. There are quite a few achievements that you will unlock playing as him. Also, if you start as Dominic from the beginning, you will not gain any less points then you would as the main character and it will save you a lot of time.