Translation:I look in the kitchen, but the chef is not there.
Yes, I think this one is just incorrect. The "move in into" I could imagine saying, since "move in" has a pretty different meaning than just "move". I'd certainly never say "look in into".
The chef usually is the alpha cook, no? The leader of the bunch. That would be expressed with főszakács - lit. "head cook".
" i look into the kitchen but there is no cook there" is wrong : ( could someone translate it into hungarian
Benézek a konyhába, de ott nincs szakács.
The main difference between the given sentence and your version is that in the given sentence they are looking for a certain cook ("a szakács"), but you were asking about a cook in general.
If you are talking about where something is, you need to include a form of van. And "nem van" becomes nincs.