Indeed, it sounds strange, but if you provided more context we could help you more.
"Chętnie" translates into happily. It comes from the word "chęć" which is will/willingness/desire.
"Chętnie chciałbym" translates into "I would willingly want/like to", which is a bit of a pleonasm like "black darkness".
You may want to use "chętnie":
*with perfective verbs e.g Chętnie to dla ciebie zrobię - I will hapily do this for you/I am happy to do that for you.
*with verbs other that "chcieć" in a conditional form eg. Chętnie zrobiłbym to dla ciebie, ale nie mogę. - I would happily do that for you, but I can't./I would be happy to do that for you, but I can't.
*as an answer to a kind request eg.
- Zrobisz to dla mnie?
- (Bardzo) Chętnie.
- Will you do that for me?
- Happily/With pleasure.