This is Exercise 8-1 in Milne's book of algebraic groups:
Let $k$ be a field and $G$ be an algebraic group over $k$ (not necessarily smooth or connected). Let $Z$ be the center of $G$, then $G/Z$ is affine if $G$ is connected or $[G,G]$ is affine.
The connected case is already done in the book; we use the action of conjugation and we take nilpotent neighbourhoods of $1$.
I'm having problems with the other case. Here is my idea:
It suffices to show that $G/Z(G)^0$ is affine. Let $G^0$ be the connected component of $G$, then $G/Z(G^0)$ is affine. From the exact sequence $$ e\longrightarrow Z(G^0)/Z(G)^0\longrightarrow G/Z(G)^0\longrightarrow G/Z(G^0)\longrightarrow e $$ we are reduced to show that $Z(G^0)/Z(G)^0$ is affine. This is equivalent to show that $Z(G)^0$ contains the anti-affine subscheme $Z(G^0)_{ant}$ of $Z(G^0)$. On the other hand, we have that $[G,Z(G^0)_{ant}]\subset [G,G]\cap Z(G^0)_{ant}$ is affine. So the exercise holds if for instance $Z(G^0)_{ant}$ is an abelian variety, which is not necessarily true. I do not know how to proceed, or even if this is the good way to solve the problem. I would really appreciate any solution or hint.