It's not a bad effort. The usual usage is: "Someone or something + brings out the best in + a person." We might then add more sentences to explain how this happens. e.g. Jane really brings out the best in John. Since they have been together, he has been so much kinder and gentler. I suppose that you could say: "I try to bring out the best in myself" but I never hear this. It sounds unusual. We have two other expressions: to show yourself at your best, or to show your best side. I think that these might be better in your context.