[There will probably be many things I missed]

After having played Outsider for a bit I thought it might be helpful to post my findings of how to play him in here. 

So you basically just want to use his Skill [his “E” ability] a couple of times into an enemy wave and then use his “Ultimate” [his “Q” ability] which consumes his HP based on the number of times you used the Skill, which then gives him increased attack speed and an Ultra Shield based on the HP consumed. The Ultra Shield can be increased by receiving healing which converts the HP healed into the Ultra Shield. He also enters the Relentless Hunt State which gives him stun immunity and a Resolve boost (Resolve = more damage on low health). 

When in the Relentless Hunt state there will also be a small timer next to his HP bar indicating how long he will be in that state (IMPORTANT: Using his skill while the timer is active will leave the Relentless Hunt state!) 

That is all I could find out in the couple hours of playing him. The information in this post is probably all over the place but this is my first time doing this kind of thing so I really, really hope it was helpful to some of you!