He does utilize the power of the Saiyans... That's why when he was blew himself up and then came back he had a golden aura around him, people considered this form Super Perfect Cell..
Take a look at this picture..
Notice that he has the Golden Aura and the electricity surrounding him... This is only commonly found on Super Saiyan 2 form that Gohan had at the time.
Also no Cell cannot become a Great Ape because he is does not have a Saiyan tail... Only Saiyans with tails can transform into the Great Ape and this in turn means he cannot become Super Saiyan 4 (which is not a canon form anyway).
Felt this needed an update after seeing people up vote this a few times.
If you think about it in the terms of hybrids, in all technicality Cell is a Saiyan Hybrid because he has DNA from multiple Saiyans. If just having the DNA is enough to go Super Saiyan, comparing him to people like Gohan, Trunks, Goten, etc who are not full Saiyans then we can assume he can use the power of a Super Saiyan.
Edit: With the recent information given by Akira Toriyama in regards to S. Cells being necessary to go Super Saiyan, we are left with the question of does Cell actually have enough S-Cells to go Super Saiyan? I believe with the previous information about the similar aura and electricity points to him having S-Cells. Another thing to note is that he has the DNA and cells from people like Vegeta, Goku, etc.
One last thing to mention is that Cell also transformed his body similar to Trunk's Grade 3 Super Saiyan, the bulky form.
Overall the answer is Cell does have his own version of Super Saiyan but as far as we know it's unsure of whether or not he can go Ape, but I'd say he probably can't.