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It sounds as if the issue does not originate from the admittedly vague definition of an unaligned character, but from the need your player to justify his (evil) actions through that ambiguity.

Kaz Magnussion, Holy Zealot of the Raven Queen, is an unaligned Human Fighter / Warpriest. He lives his life by a strict code and does not concern himself with the philosophic mores of good or evil, nor does he participate in the secular matters of kings and governors. He believes the members of his party are key players in a divine prophecy which is too great for his mind to fathom, and so travels with them in an attempt to understand the fragments of the cosmic mystery his goddess placed in his consciousness. He protects them from harm with his martial prowess for this reason, though never uses his powers to revive them from the brink of death as he sees the hand of the Raven Queen as omnipresent, guiding them toward their destinies. When innocents die around him he prays that they reach the afterlife that is denied to him but feels no remorse for their deaths, any more than he does for the foes that die under his blade. His true devotion remains in the worship of his Queen and the belief in fate, mystery, and death.

I disagree that unaligned characters should be morally apathetic, as the PHB implies. Some believe in the existence of something greater than good or evil, law or chaos, as some have already stated. A personal code, the search for truth, the restoration of a lost empire or bloodline.

As for your wizard, killing someone because they're annoying or you want their stuff...in my game, that's evil. Especially if he's metagamin' and realizes it's an NPC and not, say, some incredibly powerful agent for the king/power structure come to contact him in secrecy. In which case he should get a level 27 rogue with full action points all over his decrepit ass.

Alignments are purely fluff in 4e. Why does everyone get so excited about "My unaligned players are doing [insert something abiguously evil]!" Alignments are just a reminder written on the character sheet to the player saying "this is how I intended to play the character." That's it. It never even has to come to the DM's attention. How NPC's see them (since there are no longer any alignment discerning abilities) depends entirely upon the characters' actions actions. A DM doesn't have to know their alignment to see what they're doing.

A question I'd like to ask: How cooperative are your NPCs generaly ?
Because there could be a cause in that. If they are helpfull and only ask justified payment or reason for services then I have not much to add.

But if they are all:
- Prophets or sages who can't be bothered to give a straight answer
- Authorities who refuse to believe the PCs warning about the latest threat or defending against a wrongfull accusation
- People who send them on annoying fetch quests in exchange for a (to them) small job or item or information.
- Mistrusting civilians
- Merchants who insist on players paying full price despite the fact the monsters are literaly just outside about to butcher every living being in sight.

Then I'd also seriously consider going into what Mass Effect refers to as the Renegade path. Maybe not so much on table RPGs, but CRPGs tend to have NPCs treat characters as if they were suckers ready to do any dirty job for a small service. Maybe your player has been playing one of these games a little too much and went: "That's it. I can't stand it anymore".