I was curious about this myself, as I'm a bit of an achievement collector. Luckily, I stumbled across a useful list over at Xbox360Achievements!
They are as follows:
- Shock Jockey and Possession
- Shock Jockey and Murder of Crows
- Devil's Kiss and Possession
- Devil's Kiss and Murder of Crows
- Undertow and Shock Jockey
- Bucking Bronco and Devil's Kiss
- Bucking Bronco and Charge
- Devil's Kiss and Charge
To perform these combos, simply hit them with one Vigor, switch to the second Vigor, and hit them with that one. It will count as long as they're still under the effect of the first Vigor.
These worked for me when it came to trying for the achievement. The timing can be kind of tricky on a couple of the combos, especially if you accidentally kill your target when you're trying to get it!
As for the effects of these combinations, some are pretty minor (increased damage and prolonged vulernability), but there are a few that have some decent bonuses. Devil's Kiss and Murder of Crows gives you flaming crows, Charging a person under the effects of Devil's Kiss causes a wider spread of embers, and Charging an enemy lifted by Bucking Bronco makes it likelier the enemy will no longer set foot on Columbia. Ever.
This video goes into slightly more detail, but a lot of what they mention were things that I honestly didn't notice helping all that much.