Step 1: Compute the fundamental group of the link complement. There's a bunch of ways to do this. You'll want a procedure where the output presentation makes it easy to identify the meridians and longitudes of the link. The Wirtinger presentation is quite good for this.
Step 2: Dehn filling is attachment of $S^1 \times D^2$'s to the boundary. You can think of that as the attachment of a $D^2$ followed by a $D^3$. Attaching $D^3$ does not change $\pi_1$ so you're left with only the $D^2$ attachment. By SvK, this amounts to adding a relator to your presentation from step 1.
I'm pretty sure this is in Rolfsen's textbook. Are you reading that?