A transsexual woman is:
a male-to-female (MTF) transsexual or transgender person and the term trans woman is preferred by some individuals over various medical terms.
Because a transsexual woman would view herself as of the female gender (though of the male sex), the term "gay" applies as it would to any other woman. That is, a gay transsexual woman is a gay woman, who happened to be born as a man.
As for your original questions --
Is it a man who became a woman, then turned gay and started liking other women?
Or is it a woman who likes other women (thus is gay), and then turned into a man?
You cannot, simply by the English language, tell which came first. Sexuality and gender identity are fluid and personal, and standard phrases (such as "gay transsexual woman") will not be able to tell you whether the individual first realized that they were gay, or first realized that they gender-identified as a woman. Only the individual, if they feel comfortable telling you this, will be able to explain.
What if you changed the word order, and got a "transsexual gay woman"? Then breaking it down, you have a gay woman. This woman is also a transsexual. But you don't know which part came first still--were they a transsexual woman who later realized she was gay? Maybe. Perhaps they just feel more comfortable identifying as a gay woman, who is transsexual. This is their identity, but the term does not capture which came first.