I’m making a post about this because someone asked me about this via Flickr and I think I should make the answer more widely available. I learned about how to dye Barbie’s hair from this web page written by Carlota titled Gonna Dye That Blonde Right Out of Bab’s Hair! It’s a free tutorial that’ll tell you everything you’ll need about dyeing Barbie’s hair.
There are two further things I’ll add. One is that this method works only on platinum blonde Barbies. If you have a doll with red, brown, black, or any other colored hair, you’re out of luck because this method won’t work so well and you could end up with a doll with a yucky hair color that you didn’t intend.
Two, because of the dyeing process and the original platinum blonde color, the resulting new hair color will be a shade lighter than the acrylic paint color so, when deciding on a particular hair color, try to visualize the color at a shade lighter. If you want a really intense color of—let’s say—blue, you’ll need to use a dark shade of blue paint (such as phthalocyanine blue).
UPDATE (February 16, 2015): Unfortunately that page has since been taken off-line. The good news is that it has been preserved on the Internet Archive so one can continue learning how to dye the hair of Barbie (or any other platinum blonde-haired dolls for that matter).
UPDATE (October 3, 2016): For those of you who don’t like using the Internet Archive, a user on Tumblr known as dolljunk has reposted the original tutorial on his Tumblr blog.