This is not an example of a double negative. Whilst ther are cleary two negatives in the sentence, they refer to different things; the implication is "(The problem is) not that I'm not happy to meet you" - i.e. the first "not" refers to the problem, and the second to the state of happiness (or otherwise!).
An example of a double negative is "I have not got nothing" where both negatives refer to the same thing (the absence or otherwise of possessions). In the case of the double negative, the literal meaning is usually the opposite of that intended by the speaker.
Incidentally, the example i've used: "I've not got nothing" is grammatically correct in Greek and means (in English) "I have nothing" - which is kind of weird coming as it does from one of the birthplaces of logic.
If "not happy" is replaced by "unhappy," it can be said as, " It's not that I'm unhappy to meet you" which, in turn, can be taken to mean, "It's not as if I'm unhappy to meet you." It may be the speaker was not exactly enthusiatic at meeting the other person, so wasn't fussed either way.