No. Currently, AliExpress does not allow direct PayPal payments.
It is not necessarily a bad thing. The main reason people use PayPal, is because it provides an additional level of protection between themselves and the seller. Aliexpress' Purchase Protection does exactly that, and their resolution team is pretty efficient at resolving disputes. You can read more about AliExpress disputes here; but, to summarize, I am happy with the way they been treating me so far.
For the instructions on how to shop on AliExpress please follow the link below.