There are three ways to say it:
1. 나는 청소해요.
2. 내가 청소해요.
3. 나 청소해요. (*)
All can be said, but if you do not have any particular nuance to add, #3 is the common way to say it.
청소하다 is the verb for "clean", so you write it as one word.
"요" makes it polite sounding, so you can just say 나 청소해 when speaking to a close friend.