To me an Unaligned character is simply who just hasn't chosen sides yet. A True Neutral character, however is someone who has: they've chosen no side and a middle road.
To a certain extent, it's almost like there's three axes: good vs. evil, law vs. chaos, dedication vs. apathy. To a certain extent, nearly all of the alignments fall towards dedication, whereas Unaligned is mostly apathetic. In this way, I see Unaligned as sort of the base alignment that all the others grow out of.
Would an Unaligned character that mostly acts good be Unaligned? It depends. Depending on how often they act "Good" it might be worth noting to the player as a GM that their alignment doesn't really fit them anymore. Again, I think it's important to let players switch alignments if their initial alignment doesn't really mesh with the way they play their character. Players should be encouraged to follow their alignment, but it shouldn't be a trap that prevents them from acting freely.
EDIT:
Not me. Ideally, I'd actually prefer there to not be alignment at all.
Which I think is unusual, because to me it felt a lot like a hybrid of the AD&D 9 alignment system and the more classic BD&D alignment that I first learned. If anything, it felt more like D&D to me than the very specificall AD&D 9 alignment ever did. Plus, the unaligned option went a long way towards "fixing"--or at least making a moot point out of--a lot of the perennial issues that folks always had with alignment.
If alignment has to be included (and honestly, it probably does) I'd rather see the 4e take on alignment than the clunky old 9-point alignment.
I think a good compromise would be to include Good, Evil, Law, Chaos, and Unaligned as the base alignments with the LG, CG, LE, CE, and TN as easily added options. That way you have a five-alignment system very similar to 4e's and BD&D's right out of the box while leaving the options from AD&D and 3e to be added in easily.
I'd also personally like the
Monster Manual or its equivalent to support all alignments right out of the box, though it's not a deal-breaker for me by any means - just a preference.