To perform a United Destroyer you need to be on a mission with at least one ally (AI or coop). It's easiest to do with a full party vs. a large Oni.
First you need to fill up the blue bar on the left side of the screen, next to your allies' health bars. This represents working as a team - for example when two of you attack or purify the same enemy/limb, you'll see some bright blue particles come out of your character and fly into the bar.
Once it's fully charged, a star will appear by your name. Move close to one or more allies (a blue line appears to show you're close enough) and press Triangle+Circle to activate the United Destroyer. Much like the normal Destroyer, it will kill small enemies and sever limbs from large ones. If you have more than one ally in range upon activation, you will do more damage and sever more limbs.
In coop, only the player who contributed most to the teamwork bar will receive the star and be able to activate the United Destroyer.