After calculating the eigenvalues. I got 0 as one of my 2 eigenvalues. Hence, one of the singular values of my matrix is 0. So $\sigma_1 = 0$
But then when Im trying to find U: $$u_1 = \frac{1}{\sigma_1} Av_1$$
What do I do? $\frac{1}{\lambda_1} $ is undefined.
The correct form of this equation (since we can't divide by $\sigma_1$) is $$ \sigma_1 u_1 = Av_1 $$ Since $\sigma_1=0$, we are free to select $u_1$. Keep in mind, however, that $u_1$ must be chosen to be length $1$, and orthogonal to all other $u_k$.
So, what you do is as follows: first, find all the other vectors $u_k$. Then, select any vectors for $\sigma=0$ such that, all together, the $u_k$ form an orthonormal basis.
You may want to use Gram Schmidt in order to do this.