I think you only rank up after the season ends.
Kotori is probably the key component of my team. I use her for her AoE heal, revive and most importantly her AoE polymorph which I maxed out to increase the probability of CCing the enemy team and I usually open with this move. I also have Lucia as my primary damage dealer and Shizuru as secondary damage dealer. I've equipped Lucia with an AoE paralysis and Shizuru with AoE break just in case Kotori's polymorph completely whiffs (sometimes they all whiff which is really unlucky but Kotori's revives usually keep me in the match)
I find the CPU always focuses the lowest health target so it always tries to kill Shizuru first which happens to suit my team just fine as Kotori can heal her, while she has a pretty strong lifesteal attack that uses odd ignis and Lucia is free to slaughter the other team.
Edit: I think the multiplier may depend on how quickly you defeat the first enemy (just chose a real easy opponent that had only 1 character and wiped them in a single turn - got a 1.5x multiplier)