The first real question you need to answer is - What is your budget??
Next step is you need to figure out is what do you want to get out of your system. If you truly want to max crysis 3 at 1080p (1600 would be a different story) then you're going to have to go pretty high end. Don't listen to console users, they have no concept of true PC gaming - There is a night and day difference between console 720p 30fps compared to PC 1080p, maxed out at 1080p - anyone who says otherwise clearly does not own a gaming PC.
Next, Build your own PC, don't buy a pre-built system - you can get much higher end parts for much less, and now a days building a pc is as easy as putting together legos. You can get step by step instructions on putting together virtually any pc configuration by youtube.
If you truly want to max crysis 3 I'm going to assume you want to play crysis 3 maxed out with SMAA 2x at 1080p @ 60fps. some people can tolerate 30fps, but once you play at >60fps you never want to go below that.
1. Case - don't skimp on a good case - having a large case with good airflow is just as important as investing in the cooling itself. Personally I have a corsair HAF-X and I love it. invest $75-150 on a case.
2. Monitor - having the best pc hardware will only get you as far as your monitor will display it. I recommend a 120hz monitor, but 60hz is ok. But remember even if your hardware outputs > 60fps, if you have a 60hz monitor, you will never see anything better than 60fps. Get a monitor with a fast response time (<2ms), preferably 1ms response. Acer actually makes very good monitors for the money. Personally I'm using the Asus 27" 120hz 3d monitor. I actually do NOT recommend the monitor I have because the backlighting is terrible - things that are supposed to look black actually look grey and washed out. My Acer 120hz 3d monitor is actually much better for 2d graphics. (which brings me to my next point - I don't recommend 3d - I have it but never use it, its gimmicky and the colors are terrible compared to 2d mode). Invest at least $175 or more on a good monitor. 24 or 27 inch is lovely.
3. Mobo - Nowadays you'll want at least a z77 chipset with at least 1 pci-express x16 slot (x16 for greater bandwidth). the z77 chipset will support the new ivy bridge processors. Personally I'm rocking an asus p8z77 WS mobo - but you can get cheaper z77 mobos that do basically the same thing.
4. Processor - Get a quad core ivy bridge or sandy bridge processor. I have no experience with AMD so I can't comment on those, but I really love my intel I7-3770k processor. Also get a "k" series or equivalent processor which will allow over clocking. And again if you over clock a processor at all, you will need to invest at least something with some aftermarket cooling. Personally I have everything water cooled - but this is not necessary - good air cooling with a good case will get the job done
5. GPU - you'll need a gtx 680 or 770 or equivalent if you want to max C3, no getting around that. Expect to spend $350 on a gpu or more if you want to max c3. I personally have a gtx 690, but I wish I would have just went with 2 680's in SLI because I'm currently not taking advantage of my dual x16 pci slots on my motherboard.
6. Ram - any 8gb ddr3 ram will do, you can get pretty cheap ram now - $50-$75.
7. PSU - with a high end system like this you'll need at least 650W psu - 850W psu if you want to SLI anything. expect to spend $150 on a decent psu.
8. SSD - With a nice system you'll want to complement it with a fast SSD for the OS and games. its very nice only taking 10 seconds to boot or load a level. you can get a decent 120gb ssd for $110. you could spend up to $400 if you wanted to though.
All in all expect to spend ~ 2.5k for a system capable of maxing crysis 3. If you do it right, its all worth it in the end. Heres a picture of my baby