The elemental finish moves work the same way as regular finish attacks. You must charge up your combo by doing combination attacks, and then when the bar is full and sparkling (yellow, not red), you must trigger the attack in time. For it to be one of the elemental finishes, you must have first used a power and trigger it on an enemy still in that specific status. You will not have all of these powers right away, but you will eventually get them all.
For the Purification God achievement: To use the purification power, hold RB and press A. This power will stun all enemies around the Majin, as well as charge your weapon for a purification attack as long as you are also close.
I wouldn't worry about getting these until the end of the game as there is a save point in Arkela Castle - Hall with around 8 enemies. You can keep killing these enemies using all the different elemental attacks, return to the save point and change costume then press B. The enemies will respawn and you will be able to repeat. This will also help you max out the required levels of both characters.
NOTE: Make sure to save regularly incase you die as this will reset your XP to your last save.