It can definitely be self-taught!

A promising path would be to self-teach as much as possible and then apply for an MS program in bioinformatics or something similar.

Apart from the usual advice for teaching yourself programming, you can practice bioinformatic-specific skills at the wonderful Rosalind Project [0].

[0] http://rosalind.info/about/

The programming part can definitely be self-taught, that's basically how virtually everybody actually goes about learning programming successfully.

You already know some biology, so you can probably say more about that side of the equation.