Who was Luke writing to in Luke chapter 1?

The two letters are addressed to Theophilos (normally transliterated as Theophilus). The precise identity of Theophilus is uncertain. A Word that Luke uses to describe him could suggest that he was a member of Roman nobility or civic administration.

On the other hand, the Word "Theophilos" means either "beloved of God" or "God-lover", so it is at least possible that the gospel was not directed to any specific individual, but rather to each individual believer.

Luke doesn't give us enough information to conclusively know the identity of the recipient of his text.