Necrons are a fairly solid codex right now. They don't have any glaring issues like some of the older codexes do. They come with very impressive shooting and decent resilience. I would say they are a top-tier army this edition, but they aren't all-powerful.
They are not the greatest army in melee. They have units that can be good at melee(Overlords, Wraiths, C'Tan, several of the elite choices), but the core of the army(Warriors and Immortals) won't stand up very well in assault. And those units that are good tend to be fairly expensive. Lynchgaurd and Pretorions cost the same as TH/SS Terminators but are just not quite as good. 6th edition is shaping up to have an even bigger emphasis on troops than 5th, so it's probably not wise to overpay for average quality assault units. Wraiths and Command Barge Overlords tend to be the exception. These are assault units that will find their way into most competitive Necron armies. Scarabs are a special case since they are really only effective against vehicles, but, boy, are they effective:)
While the Necrons at first glance seem very resilient, a competent opponent that knows your codex can compensate for that advantage. High strength/low AP weaponry can make short work of Necron vehicles even with their Quantum Shielding. Since Necron vehicles are opened top any penetrating AP1 hit will destroy you on a 3+. Reanimation protocols can also be overcome if your opponent focuses fire effectively. If a whole unit is knocked down, no one can get up. Plus the inclusion of precision shots could allow opponents to knock out your res-orbs at an inopportune time.
Necorns also don't have as wide a range of allies open to them as some other codexes. Though they need them to be competitive a lot less than some others. And they can ally with Grey Knighs...'nough said about that.
I had mentioned how many Necron assault units where expensive. That's a trend with many of the options in the codex. While there are many units that are cheep and effective(Warriors, Nightscythes, Annihilation Barges) there are others that can end up costing too much. The special characters are all very expensive. The rest of the HQs are decently priced, but you should be weary of giving them too many upgrades. You won't get a good return on investment if you give them more that 2-3 options. Most of the Elite choices are expensive for what they do. The efficiency of the cheaper options won't always compensate for the more costly ones. Necrons are typical of xenos armies with the amount of dead entries present in the Codex.