Hi Socko Grammatically correct is: 'I want to know what he is doing.' That is a statement not a question. If you are speaking... and you are watching him do something, and you're talking to your friend, you could say "I want to know .....[pause] what is he doing?" which is basically two sentences, and ends in a question. Sometimes people write in a casual way to imitate how they speak, such as in email or on Instant Messenger conversations, and in that context this is perfectly understandable. But technically it's not grammatically correct. ---- Added a bit extra: Your question makes me think of a song that shows the sentence construction you're asking about. A pop song song called "Pure Energy" has a refrain (the part of the song that repeats) that says: "I wanna know what you're thinking there are some things you can't hide I wanna know what you're feeling tell me what's on your mind. " ("wanna" is spoken slang for "want to") You can hear it yourself at this video http://www.youtube.com/watch?v=lsHdea5SRjU (the refrain starts at the 1:24 mark). Perhaps hearing this song will help you.

“I want to know what he is doing." is correct because word "what" is here as a relative pronoun not question word.