Yes, the dual of $SL_2$ is $PGL_2$.
But you're not going down the right track with $PSL_2$. The problem with $PSL_2$ is that it's not a variety at all! You can quotient out the variety $SL_2$ by the subgroup $\pm1$ but the quotient is the variety $PGL_2$ (recall that quotients in the category of sheaves (for these are really fppf sheaves) don't have to be surjective on global sections, so the statement that there's a surjection $SL_2\to PGL_2$ does not imply that the induced map $SL_2(\mathbf{Q})\to PGL_2(\mathbf{Q})$ is a surjection).
The problem with $PSL_2$ is that it is a functor from, say, rings to groups, but it's not a representable one, so in particular it's not an algebraic group. If you like, you can imagine $PSL_2$ as a presheaf quotient, and $PGL_2$ as the associated (representable) sheaf.