"I am Welsh."
You can also say:
* "I'm Welsh." (shortened version)
* "I'm from Wales." (more specific)
* "Rwy'n Gymro" (in Welsh, meaning "I am a Welshman")
* "Rwy'n Gymraes" (in Welsh, meaning "I am a Welshwoman")
The best way to say it depends on the context and who you're talking to.