What is a single word (or two words) that you would use to describe a person who is trying to come out of his troubled past (like memories of loss/grieving)?
Let me start by saying that "troubled past" doesn't describe loss or grieving. Usually a person with a troubled past has had difficulties with the law, substance abuse, difficult childhood, or the like in his past.
I will assume that you are misusing the term troubled past and are actually referring to a person who is attempting to recover from a personal tragedy of some sort. Looking up "recovering" in a thesaurus gives several alternatives: getting over, picking up, perking up, picking up the pieces, feeling better, getting his life back together, coming along. In all these cases except getting over, you can say "He's had a personal tragedy, but he's [picking up/perking up/etc] now." With getting over, you'd have to say "he's getting over it now."
You may also say "He's recovering from a personal tragedy" or "He's getting over a personal tragedy." The others would work a bit differently: "He's getting his life back together after a personal tragedy", for example. You can use after in the same way with most of the others (you can use it with recover as well), but I wouldn't say "picking up after a personal tragedy" because it can easily be confused with the more literal sense of picking up: "picking up after a party" would mean gathering the trash. If I were going to use picking up, I would say "picking up after having had a personal tragedy". This sounds hopelessly stilted to me, so I wouldn't use it.