Is this build compatible?

[SizedMouse9]

I'm trying to upgrade my computer without breaking the bank. All I want is a graphics upgrade but i have to upgrade my CPU as well. The only things I want to keep is my storage, ram, and motherboard but everthing else im flexible with

The computer I have now
https://www.bestbuy.com/site/cyberpowerpc-gamer-ultra-desktop-amd-fx-series-16gb-memory-2tb-hard-drive-black-blue/4592900.p?skuId=4592900

CPU: AMD FX-8320
GPU: MSI Radeon R7 250 1GB
RAM:A-Data 16gb 1600 MHz DDR3
MOBO:Gigabyte G-78LMT-USB3 6.0
Storage:Toshiba D101ACA200 (2 tb HDD 7200 RPM)
Dimensions: Height 17 in, Width 7.6 in, Depth 19.75 in

Here are the new parts I want - https://pcpartpicker.com/list/2TXgZR
SeaSonic - EVO Edition 620W 80+ Bronze Certified Fully-Modular ATX Power Supply
MSI - GeForce GTX 1080 8GB
Cooler Master - Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler
AMD - FX-8370 4GHz 8-Core Processor

Will this build work? I want to keep some parts so I will have a decent computer and can upgrade it in the future while keeping the goo GPU. Will I have to buy a new case? I dont know if the GPU will fit the current case I have and I dont want to spend more than $50 on a new case.

If you have any suggestions or comments please let me know. The computer I have now is trash and can barely run any game and I am in desperate need of upgrading it without spending tons of money on extra parts I dont need. How long will this build last because I probably wont be able to fully upgrade for 6-12 months.

 

[WildCard999]

Waste of money going from the 8320 to the 8370.

For $1000 I'd get this...

PCPartPicker part list / Price breakdown by merchant

CPU: AMD - Ryzen 5 2600 3.4GHz 6-Core Processor ($164.99 @ Amazon)
Motherboard: ASRock - B450 Pro4 ATX AM4 Motherboard ($69.99 @ Newegg)
Memory: G.Skill - Ripjaws V Series 16GB (2 x 8GB) DDR4-3200 Memory ($149.89 @ OutletPC)
Storage: Crucial - MX500 500GB M.2-2280 Solid State Drive ($89.99 @ Adorama)
Video Card: EVGA - GeForce GTX 1070 8GB FTW2 Gaming iCX Video Card ($340.00 @ Newegg)
Case: Fractal Design - Focus G (White) ATX Mid Tower Case ($49.99 @ Amazon)
Power Supply: EVGA - SuperNOVA G3 650W 80+ Gold Certified Fully-Modular ATX Power Supply ($75.88 @ OutletPC)
Total: $940.73
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-09-15 15:22 EDT-0400

It's much more balanced over a 8320/GTX 1080 and will game much better due to the better CPU.
http://cpu.userbenchmark.com/Compare/AMD-Ryzen-5-2600-vs-AMD-FX-8320/3955vs1983