It might be The Mind Thing from Fredric Brown (published in 1961).
The being (The Mind Thing) has no arms or legs, and looks much like a tortoise shell minus the head and legs. But he possesses a terrible power. The Mind Thing can enter the mind (and thus control the body) of any sleeping creature. The catch is, it can only exit the possession of such a creature by killing it, i.e. by causing it to commit suicide.
The story focuses on the efforts of The Mind Thing to control the necessary humans to enable it to return to its home world. Oh, and by the way, The Mind Thing has determined that human beings are ideal hosts for its race, so if it succeeds in returning, eventually its race will raid the earth for its slave potential.
from an Amazon review.
The Mind Thing is teleported to Earth due to its crimes, but by its people's laws if it can find its way back it will be pardoned. The Mind Thing uses some of his hosts to hide its body and also to prepare the food for it (needs to be soaked in nutritive fluid). It also gathers the knowledge of the host it possesses, and eventually goes trying to possess smarter animals or smarter human individuals. All this fits with your description of the story.
Highly recommendable reading, as almost everything from the pen of Fredric Brown.