Destructive component...meaning he can potentially hurt others and/or himself with his power? If so, I have a rather cliche answer to that. If Alexander wields holy power, and if he is currently surrounded by demons, it could be safe to assume that the same holy powers that can heal himself/other humans can easily kill an unholy creature, a.k.a. a demon. This could be a good reason why Lilith would fear him--he could easily kill her if he knew he could and if he so wanted to. 

However, this kinda kills your idea of Alexander healing Lilith...unless he can heal a demon in moderation. Like, a little bit of his power could heal Lilith, but using a lot of it at once could kill her. It would be like the difference between the sunlight warming one's skin and burning the ever-living hell out of it, lol. For Lilith to allow Alexander to heal her, she would have to trust him enough to know that he won't kill her with it. For Alexander to heal Lilith, he would have to have great control over his power and trust her enough to not want to kill her.

This, however, doesn't solve the issue of how to initialize trust between them. Well, already, Alexander has made the connection that his treatment as Lilith's captive is better than what his treatment would have been as a captive in Afghanistan (though that could change, lol). If she continues to treat him with reasonable civility, he'll be less guarded. However, I don't think he could begin to trust Lilith until he learns what she's all about. He'll need to discover what other demons are trying to do (run amok in the world) and that Lilith is trying to stop that. He needs to realize that her cause is actually a good thing. Until then, his focus will be on developing a plan for escape regardless of her kindness.

So, I think it's gonna be a little while before we start to see some real character development between our two characters. Alexander needs to learn what this version of Hell is all about first :)  

Hope that helps with ideas? Do you prefer the destructive side of Xander's power to be more original/less cliche? If so, I can scrape together some ideas...