Can $\mathrm{PGL}_2$ be viewed as an affine algebraic group? I was just wondering whether or not it is possible to view $\mathrm{PGL}_2$ as an affine algebraic group. 

Yes, $PGL(V)$ is a linear (affine) algebra group. For more on this you can see these notes:
https://www.ma.utexas.edu/users/allcock/lag/lag14.pdf
See Theorem 7.1. Remember that $PGL(V)$ is defined as a quotient $GL(V) / \mathbb{G}_m$.
