Here's the situation: my PC's alignment was neutral, but I don't feel like it matched the personality I wanted him to have. Can I change my character's alignment and, as consequence, his deity?
People change over time as can alignments shift to nearby points on the good/evil law/chaos axes. The only major snafu that ever comes into play is when one goes between radically different points multiple times.
I cannot recall the source but deities and other aligned creatures tend to get annoyed if you keep bouncing all over the alignment chart. Especially diametrically opposed forces like good and evil. Its one thing to fall from grace and then later try and redeem yourself. Its another to consider alignment as a temporary resting point between adventures. But that bit tends to be more on the lines of DM's discretion than a hard and fast rule.
Also of note, only certain classes and certain class features are bound by alignment. Once again, DM discretion on house-ruling such things.
Overall, your alignment may change but don't make it an issue for the rest of the group by causing problems (basically the rule of DBAD).
4e really doesn't care. Matters of deities and alignments are really left entirely up to the DM. Talk to your DM about this and he may develop some narrative consequences to the change, however, there are no mechanical means nor consequences to make the change.
Also, as Oblivious Sage points out, you can totally change your alignment without changing your deity (you're explicitly allowed to not be of the same alignment as your deity unless you're her Invoker or Paladin). Though again, there may well be narrative consequences. You can also change your deity without changing your alignments...again possible narrative consequences.
The only hiccup here is if you have a character feature (namely a feat or paragon path) with your previous deity as a pre-req. You would need to retrain these first. That's the only potential RAW mechanical impact (And I would generally advice against creating additional ones). However, again, there may be narrative impacts, talk to your GM.