Oi mate, i build systems in my freetime and often resort to used parts, maybe i can do something for you here ^^
Lets get started with the most basic stuff, Mobo, CPU, RAM and of course, your graphics card.
Your current system sports a Notebook CPU, the A8-7410 was launched in 2015, so its a few years old and the 4GB of RAM arent doing you a favour either.
The route i would suggest is going with cheap and used I5s and I7s, you will get decent performance even for modern titels with a used I5-6600, you can pick those up for around 150-200 bucks.
Throw in a Z270 or Z370 motherboard for 130-180 bucks for good measure and top it up with 8GB of DDR4 RAM for 80 bucks a pop brand spanking new. You can go with used here, but not many people sell DDR4 as it is the current generation.
If you need a cooler for your CPU as well, take the most basic Hyper212 for 20 bucks brand new.
your system needs power, since youre on a tight budget, modular wont be the leading thing here, so i suggest a nice 500W PSU, look into some from maybe Corsair for like 80 bucks brand spanking new. You want to take close attention if you buy used here since your powersupply affects every other component in your system, so dont cheap out here.
As for mass storage, you can grab the HDD out of your old rig then, i read it has a 1TB drive in it.
As for the case, just take a cheap (maybe even used) case that supports ATX as your formfactor or mini/micro-ATX. You can get cheaper mobos in the smaller mini and micro formfactor, but you do have to watchout since your cooling solutions will be more limited and the choice of your GPU is limited as well.
Were now at:
Case: like, 60$-100$
that leads us to a grand total of: 540$-680$
The rest of that budget can go into your GPU.
you can go risky and buy used mining GPUs, look for those who specifically say that they were undervolted, never ran hotter than 70°C and who got taken care of nicely. If youre not sure about it, leave the deal and hunt for another.
As for otherwise, get a used GTX 970, 1050ti, 1060 or 1070 (yes, those too, ive seen ads for used one sat ~300$) and youre good to go.
I would always recommend to save up and build your own system, you can expand it, repair it and change parts on your own and you learn a lot from it. And its really like LEGO just with more expensive parts.
If you dont know how to build your own systems, there are tons of vids on YouTube that show you how to do it.
As for the prices, i pulled them off of ebay germany and converted the EUR straight to USD, in just a 1:1 ratio. We are on the expensive side of things then since normalle the EUR is higher in value than the USD.
If you have any questions, dont be afraid to ask, some are Elitists but most gamers who build their rig themselves are kind and share their experiences and tips gladly with you.
EDIT: I forgot to mention it, but this:
is quite the helpful tip. I got 2x 1TB SSDs for 300bucks there, shipping included. I had to pay 50 bucks import taxes, but that was because i imported it from the US of A to germany, but it is still 100-150$ cheaper than buying them fresh outta box on amazon.