Killing = dealing damage to and removing your own units. This is easily done with Skellige's Ursine Ritual with cards spawning 1p units/tokens, e.g. Harald Handsnout, or Germain Piquant. Svalblod Priest is also helpful here.
Okay, I don't know what happened, I tried several of the tips above, and nothing helped. Then after logging in again some time ago, the quest was counted as finished.
So some of the "Kills" triggered later. Sadly, I don't know which – if you're here looking for help.
"Kill 7 units" - you get it by using cards that damage enemy units (either takes them down in one shot or gradually). Not cards that "destroy" them (i.e. Vilgefortz, Brehen, Moreelse, Vanhemar, etc.). Leader abilities that destroy cards such as OH also don't count. I played a few games using various decks, killed some enemy units by damaging them or one shotting them and the quest was complete.