A couple of things to bear in mind:
- Don't queue with friends if you want this achievement done as fast as possible. It only counts people you've been grouped with randomly.
- You say you have a class that can heal so use it to reduce the time spent queuing (ideally it could tank as well).
- If you can fulfill a call to arms, do so for the http://wowpedia.org/Misty_Satchel_of_Exotic_Mysteries and the chance of extra pets (even if you get one you already have, they can be used in pet trades on other servers).
I'd suggest queuing for a random heroic as the queue times here are at least as short as each specific and there are bonus rewards. As for which is quickest (if you decide you don't want to queue random), it really depends a great deal on the speed of your tank (another good reason for tanking it yourself on a well-geared character if you had one). As a well-geared DPS, you can push on ahead of the tank without too much difficulty, but it may annoy your group. You could take a look at the CM Gold times and judge the length of each HC based on those (the correlation should be reasonably strong).