icauston - I can not find this in the PublicApi either, however I did find two alternatives:
1. The OOTB modal utilizes the following to update a user's uploaded avatar:
using (Stream inputStream = this.UploadedFile.InputStream)
user.AvatarUrl = Telligent.Evolution.Avatars.UpdateAvatar(user.UserID, inputStream);
2. Another possibility would be to use the same IUserService (in the rest namespace) that the REST endpoints use
var avatarService = Services.Get<Telligent.Evolution.Rest.Resources.Users.UserService>();
However I am not sure about the ramifications of using this outside of a REST call, and looking closer, this call uses the same underlying code as in option #1.
Have you thought about just calling the REST endpoint to update the avatar from within your C# class? It may be more of a hassle, but would be supported.