Just enter your new name, choose your .fch file and hit the rename button!
Valheim stores your playerprofile and playerdata in a .fch file. Everything about your player is inside this binary file. It also contains map data for each world you played, and because of that the file can get relatively big. This site can only handle files up to 128MB for now (and only if your upload speed is sufficient to upload it in 60 seconds), sorry. Nothing of your .fch is stored on this server.
Everything you have build before renaming your character will still show the old name (i.e. crafts, beds, etc.).
Your local character files are located at:
Enjoy the awesome game and have fun playing Valheim!