Can I migrate my hard drive without changing Windows?

No, you're going to need to reinstall.

If Windows is OEM then you would need to install it again otherwise you can try it with the new system. I've non OEM Windows and after buying new mobo, previous installation of Windows worked fine. However, drivers still need to be installed in this case.

A) If this system is a Dell, Gateway, etc. OEM computer with W7 or earlier NO it can NOT migrate to a new system. The OS is locked to the specific hardware sold with the original system (Mobo, CPU and sometimes even the stupid case! - THANKS Emachine!) .
B) TERRIBLE 'upgrade'. APUs, even with a GPU are NOT gamer machines and won't be a 'serious' improvement, infact as you get the newer (2010 and forward titles) you will experiance more and more issues as the APU is made for the other 90% of the crowd out there (get on Facebook, check email, watch Youtube).
C) the AMD Radeon HD 6670 was considered for very low graphics settings 3 years ago (http://www.pcmag.com/article2/0,2817,2385800,00.asp) so this isn't even a playable card for todays gaming, even if trying for low graphics.

Honestly, the picked parts, it looks like your trying to build a $200-$300 PC gamer's rig, which isn't possible or playable (playable is normally over 45FPS, 1080p Screen on High Graphics settings). Your looking at 1,024 by 768 video (WAY YUCK) and gettings 10s to maybe 20s depending on the game <<EDIT: These facts come from the above review and testing, please see link above>>, if not as low as single digits (say BF4 64man map).

If your going to stick to this price range GO CONSOLE, it is MUCH better VALUE, and PERFORMANCE for only $149 (PS3 or Xbox 360) that still plays modern titles just as well as old titles, and you never need to worry 'if my console can play this game', as you do need to worry with PCs.

A 'normal' playable desktop PC from scratch is about $700 or more, with a decent PSU to power the GPU, and a i5 / FX8xxx CPU. There is plenty of deals to be had, like a $349 i5 desktop PC (slickdeals.net), then buy the GPU of choice ($139 for a GeForce GTX 750 Ti http://pcpartpicker.com/part/pny-video-card-vcggtx750t2xpboc which needs only a 300W PSU! ) you can wiggle down that cost and get some excellent performance now. Of course in 2-3 years you have to replace the GPU anyway as it won't be able to hold up to the current 2017-2018 titles, then at 5 years buy a whole new computer again as your i5 can't hold up to the current demands in 2019-2020, which is why most people go consoles since they only got to buy one every 10 years.

Yes you can - http://www.sevenforums.com/tutorials/135077-windows-7-installation-transfer-new-computer.html

I just posted that here just in case anyone else needed it. I had needed it and found that other after a while link but I had came here first.