How can I multiply a binary representation by ten using logic gates?
I assume that the task is to compute $mul(10, a)= 10a$. You don't need to do multiplication. A single binary adder is enough since $$10a = 2^3a + 2a$$
meaning you add one-time left-shifted $a$ to 3-time left-shifted $a$. 
For general multiplication $mul(x,y)$ please see this article.