For starters, you will need to acquire all vigors. You will get 4 free samples when you start Clash in the Clounds DLC. Once you save up enough money you can purchase the rest of the vigors. I recommend you purchase the upgrades for each vigor as well. However, this is not necessary.
Now here are the 8 different combination moves:
• Possession and Shock Jockey
• Possession and Devil's Kiss
• Devil's Kiss and Charge
• Murder of Crows and Devil's Kiss
• Murder of Crows and Shock Jockey
• Bucking Bronco and Devil's Kiss
• Bucking Bronco and Charge
• Undertow and Shock Jockey
You will want to load up Raven's Dome (preferably waves 1-3) since they are the easiest to grind for these. I personally would have a hard time explaining in writing what you need to do for each scenario. I will link in the video I used for a couple of the harder combinations. Credit to VidGamiacUnlocked for his video and stellar gameplay to help a moron like me finish my last 2 combos!