The technical requirements for most PP software are fairly modest (eg around 2Gb of RAM I think) so what you can "get away with" depends to a large degree on your level of patience
. [On my old PC I was running LR3 on a bit over 1 Gb of RAM - it worked ok, just meant when I moved a slider I had to wait for a second or two for the display to catch up - useable, but not ideal obviously.]
I don't know much about laptops but would suggest that if you don't need the portability then a desktop offers the best solution. This is mainly due to the ease of expansion - you can buy what you need now and add more RAM, disk space, etc later when needed. It also means that your next upgrade becomes cheaper as you can re-use components such as your monitor, keyboard, etc. With a desktop you can also determine the exact components you want and build it yourself (or have the shop build it for you which is usually not too expensive). If you are thinking of going this route,
whirlpool is generally a good source of info.
If you go with an Intel CPU you don't need a separate graphics card as the integrated graphics is good enough for general photo editing. With a desktop you can always add a graphics card later if you find that you need one. (I don't know if AMD CPUs have integrated graphics).
You don't need a SSD. (I think you are better off putting the money into an extra HDD, maybe external, for backup).
You can probably get away with 4Gb of RAM if the budget is tight. I don't think I use more than 4Gb with both LR and PS loaded (although I rarely check usage), but my processing is fairly simple (no HDR, no panorama stitching, no complex filters, etc). Also my images are 12 Mp - I don't know if the 24 Mp images from your D5200 would require a significant increase in memory. With a desktop you can always add more RAM later if needed (up to the max supported).
Cheers.