Not all of it, the symbiote itself is pretty strong, and Eddie Brock is as the comics say "An olympic level athlete", so that adds to his strength. But yes, MOST of Venom's powers come from Spider-Man's DNA. You see, the symbiote has a genetic memory and copies the memories, DNA, and sometimes powers of its hosts into itself, and then whoever wears that symbiote has access to those abilities.
In the case of Venom he got these pwoers from Spidey...He's immune to his spider sense, he shoots webbing (although that webbing is actually concentrated symbiote which dissolves and dies not long after it leaves the main symbiote), his ability to stick to walls (although he could have had that before, but i'll say it's a Spidey power), his basic image, the whole big white Spider thing going on, and most of his strength/speed/agility.
He also has other pwoers, like his symbiote can become any form of clothing he wishes, he can blend into anything (basically become invisible to the naked eye), he can also focus the symbiote into various weapons (IE: He could turn his hand into an axe). Venom however RARELY uses the ability to turn his symbiote into weapony type things, but he has on occasion.