There are a couple of different ways in which you can increase your power level. Most of it is quite time consuming, so don't expect to be getting up levels overnight. Here are some of the things that I have noticed contribute to your power level in Fortnite.
- Spending Skill Points
- Spending Research Points
- Assigning members to a squad. More power if the squad member is a good fit.
It seems that squad members provide the most here and assigning members that are rarer than others will give a much larger increase to your power level.