If you don't want to depend on a specific compiler, you can use the newunicodechar package together with inputenc in the following way:
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{newunicodechar}
\newunicodechar{α}{\alpha}
\begin{document}
$α$
\end{document}
Unfortunately you will have to manually declare every single unicode character that you want to use in this way. The unixode package does this for some common characters. I myself also compiled a list of those characters that I came across until now.