Component Check Please

Nice build , the PSU is weak though.

I don't like it. I don't mean to be harsh, but I will suggest some alternatives in a second post.
Processor: Intel I3-4340 3.6 GHz LGA1150 Processor BX80646I34340
Maybe, only if budget-limited. I'll find out.

Motherboard: ASUS H87I-PLUS LGA 1150 Intel H87 Mini ITX Motherboard
Ok, this I'd keep.

RAM: Corsair Vengeance 16GB (2x8GB) DDR3 1600 MHz
2x4GB is enough, and will save a nice chunk of change.

Case: BitFenix Mini-ITX Tower Case
If you're going for a small build, this case is too large. Otherwise it is okay, but I suspect there are better choices.

Power Supply:Corsair Builder Series CX 600 Watt ATX/EPS
No, unless it is the modular one. The non-modular one uses inferior Samxon capacitors that don't like heat. Especially in a small case, it isn't a good choice. It is also more power than this build needs, BUT that may change, as I go on...

Optical Drive: LG Electronics 24X SATA Super-Multi DVD Internal Rewriter
Ok. One optical drive is much like another.

Hard Drive: Seagate 1TB Solid State Hybrid Drive
Since the case has plenty of drive space, I'd probably try to get a WD Black , or try to fit a SSD into the build.

Video Card: MSI NVIDIA GeForce GT 620 2GB GDDR3
Only if you have no desire at all to play games, which seems unlikely. Your chosen case can take a much more substantial card.

Cpu Cooler: Cooler Master Hyper 212 EVO
The slightly superior Xigmatek Gaia costs less; I'd have to check currently, but $10 is typical. In a non-OC build though, you really don't need an aftermarket CPU cooler at all though, so this money can go into the graphics card or toward the SSD.

I'm at work, but I will post an alternate build that is smaller than this one and will perform better as soon as I can.

Your build would work.
I love small cases, particularly ITX.
On this build, I might do things a bit differently:
1. The i3-4340 runs cool. The Intel stock cooler is fine, no need for the EVO.
2. The GT620 only needs a 350w psu. I might use this higher quality Seasonic 380W psu. It has a 6 pin connector that can handle a card as good as a GTX660. http://www.newegg.com/Product/Product.aspx?Item=N82E16817151072
3. I have mixed thoughts about the hybrid drives. If you have room for only one drive, and need large capacity, then yes, they are good.
I would rather see a single ssd. 120gb will hold the os and a handful of games or apps. 240gb and you may never need a hard drive at all. If you ever do, it is easy to add a hard drive later.
4. The bitfenix itx is an attractive case, but it is large, the size of some M-ATX cases. 9.8 x 15.9 x 14.3
If your space in a dorm is limited, like most, consider a smaller case. For example, look at this Lian Li PC-Q03 case. It costs a bit less, and is 7.83" x 9.65" x 8.27" (W x H x D) One caveat, it will need a slim dvd drive.
http://www.newegg.com/Product/Product.aspx?Item=N82E16811112372
5. Ram is cheap, I have no problem with 16gb. If you need to shave a bit on the budget, then 8gb is ok.
Do not pay extra for the fancy vengeance heat spreaders, they are not needed.

Ok, here's a build. It focuses on small size, but performs quite well.

IN WIN BP655.300TB3L Black Steel / Plastic Mini-ITX Tower Computer Case
Model #: BP655.300TB3L
Item #: N82E16811108428
$69.99 -$10.00 Instant $59.99
The 300W PSU included in this case was reviewed recently at Hardware Insights. While it has some Samxon capacitors in it, this build won't stress it at all. There's money left over in the budget to replace it with a Seasonic unit if desired.

ASRock H87M-ITX LGA 1150 Intel H87 HDMI SATA 6Gb/s USB 3.0 Mini ITX Intel Motherboard
Model #: H87M-ITX
Item #: N82E16813157386
$103.99 -$10.00 Instant $93.99
You could use your original choice as well. I've had good results with ASRock though, and liked this one.

Western Digital Scorpio Black WD7500BPKT 750GB 7200 RPM 16MB Cache SATA 3.0Gb/s 2.5" Internal Notebook Hard Drive Bare Drive
Model #: WD7500BPKT
Item #: N82E16822136835
$84.99 $84.99
Storage drive, but reasonably quick.

SAMSUNG 840 EVO MZ-7TE120BW 2.5" 120GB SATA III TLC Internal Solid State Drive (SSD)
Model #: MZ-7TE120BW
Item #: N82E16820147247
$111.99 -$12.00 Instant $99.99
SSD Boot drive. You could get a bigger one if you wanted.

ASUS DRW-24B1ST/BLK/B/AS Black SATA 24X DVD Burner - Bulk - OEM
Model #: DRW-24B1ST/BLK/B/AS
Item #: N82E16827135204
$19.99 $19.99
One optical drive is much like another. This one has a -$3 promo code I think.

Team Vulcan 8GB (2 x 4GB) 240-Pin DDR3 SDRAM DDR3 1600 (PC3 12800) Desktop Memory (Orange Heat Spreader) Model TLAD38G1600HC9DC01
Model #: TLAD38G1600HC9DC01
Item #: N82E16820313344
$78.99 -$10.00 Instant $68.99
8GB is enough. This is fairly low-profile, which is needed in this case.

Rosewill RDRD-11004 2.5" SSD / HDD Mounting Kit for 3.5" Drive Bay
Model #: RDRD-11004
Item #: N82E16817997037
$6.99 $6.99
The SSD and HDD will both fit on this, which actually leaves a 3.5" bay free in the case.

Coboc Model SC-SATA3-10-LL-GR 10" SATA III 6Gb/s Data Cable w/Latch,UV Green
Model #: SC-SATA3-10-LL-GR
Item #: N82E16812422633
$0.53 $1.59
You'll want these shorter cables for the sake of cable management in the small case.

Intel Core i5-4430 Haswell 3.0GHz LGA 1150 84W Quad-Core Desktop Processor Intel HD Graphics BX80646I54430
Model #: BX80646I54430
Item #: N82E16819116895
$189.99 $189.99
There was easily room in the budget for an i5. The stock cooler will be sufficient.

SAPPHIRE 100357LP Radeon HD 7750 1GB 128-bit GDDR5 PCI Express 3.0 x16 HDCP Ready Low Profile Video Card
Model #: 100357LP
Item #: N82E16814202002
Mail in Rebate Card
$109.99 -$10.00 Instant $99.99
Significantly stronger than the GT620, it will allow you to play modern games on medium to high settings, depending on resolution. You'll need to turn off things like AA, but they should still look pretty good.

Subtotal: $726.50
Well under budget. You might add a decent USB wireless adapter (if you need it), or perhaps an external USB3.0 drive for backups.

I have a similar build in this case that I'm preparing to turn into my daily driver. The included PSU is indeed sufficient, and it is not loud.

An interesting mix of quality and okay parts. What is the purpose? For general use, you really don't need the GT620, gives you a few more video outputs, but not really worth it. GTX650 should be the minimum if you are buying a discrete Nvidia card. Found an MSI for $80.

PCPartPicker part list: http://pcpartpicker.com/p/1QiV1
Price breakdown by merchant: http://pcpartpicker.com/p/1QiV1/by_merchant/
Benchmarks: http://pcpartpicker.com/p/1QiV1/benchmarks/

CPU: Intel Core i5-4440 3.1GHz Quad-Core Processor ($194.99 @ Newegg)
CPU Cooler: Xigmatek GAIA SD1283 56.3 CFM CPU Cooler ($29.99 @ Newegg)
Motherboard: Asus H87I-PLUS Mini ITX LGA1150 Motherboard ($105.99 @ Newegg)
Memory: G.Skill Sniper Gaming Series 16GB (2 x 8GB) DDR3-1866 Memory ($156.99 @ Newegg)
Storage: Seagate 1TB 3.5" 7200RPM Hybrid Internal Hard Drive ($99.99 @ Newegg)
Video Card: MSI GeForce GTX 650 1GB Video Card ($79.99 @ Newegg)
Case: BitFenix Prodigy (Black) Mini ITX Tower Case ($59.99 @ Newegg)
Power Supply: SeaSonic G 550W 80 PLUS Gold Certified ATX12V / EPS12V Power Supply ($59.00 @ Newegg)
Operating System: Microsoft Windows 8 (OEM) (64-bit) ($99.99 @ Newegg)
Total: $866.92
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-10-18 13:47 EDT-0400)

If size is not as big a concern, then Eximo's selections look good too. You can get back under budget by omitting the cooler or cutting the RAM back to 8GB; actually the original build didn't list an OS, so if that's not needed, ramp up the graphics card to a GTX650Ti Boost or get the SSD and still make budget.

I generally assume students have access to Windows at a discount, either through university or directly from Microsoft, but I also like to include it in the costs just so that it is there if needed.