Move Adrenalash says add 50 damage on next turn. Does that mean every other turn or every next turn while he's active?
Adrenalash --> 50 damage
During your next turn, each of this Pokemon's attacks does 50 more damage (before applying Weakness and Resistance)
So it means exactly what is written. Every time you use this move, your next turn has an additional +50 boost for any move. The boost will continue every next turn as long as you keep using Adrenalash.
For example, here's damage output for two scenarios (Ursaring has another attack for 80 called Strength):
Case 1 Adrenalash -> 50, Adrenalash -> 100, Adrenalash -> 100 ...
Case 2 Adrenalash -> 50, Strength -> 130, Strength -> 80, Adrenalash -> 50, Adrenalash -> 100 ...
(2nd turn receives boost from first turn; for 3rd turn, and you no longer have adrenalash dmg boost because you used strength on previous turn)
So whether it is active every few turns, or every turn, is based directly on when and how you use Adrenalash. If you want to keep the boost active at all times, just keep using that move. If you stop using it, you will have to set up again with one turn of 50 dmg Adrenalash.