The GTX650ti is fairly slow so if they are running a dual core now, upgrading the system to a faster dual core or that FX6300 wont help much.
There is nothing you can do to magically make that slow GTX650ti run newer games with decent settings at over 40fps.
I am basing this on actual gaming not just 3dmark testing.
My system with a A6-3650@3.56ghz,4gb mem,win7 X64 with a GTX680 is today's min spec gaming system for 1080p gaming. adaptive vsync enabled. Metro last light redux, 1080p, very high settings, physx off, 52fps average. DOOM@50-60fps. Watch dogs 45-60fps. GTA4, 50-60fps. Assassin's CreedIII 55-60fps. Skyrim 40-60fps. Gasguzzler's extreme 64 55-60fps. GTX670 physx card, Metro last light 46fps average maxed out details, physx on.
-If they have a 1156 system look for a used 1156 i5 or the xeon version. A cpu swap would be ok. Some AM2+/am3 systems can work with a phenomIIx4. Socket 775? Some can work with faster core2quads and modded xeon 771's but do your research first.
If you manage to upgrade the existing system to a quad core, a much faster used card like a GTX680 or HD-7970 or faster would work well.
I am going to stress that a GTX680(same as GTX770/960) or HD-7970 (same as R9280x and 380X) and a quad core is the bare min for today's games. Building a system with that GTX650ti is a wasted effort.
-A used system with a sandybridge i5-2400,2500 like the SFF dell or hp could be ok. Transplant the parts into the case they have now if it's a tower case. A gaming power supply will be needed. Most come with windows 7 or 10.
-MAKE SURE IT HAS A PCIE-16 SLOT, IS A MICRO-ATX BOARD, AND IS NOT ULTRA SFF AND USES A POWER BRICK AS THOSE ARE BASED ON LAPTOP BOARDS. ALSO THERE IS ACTUALLY A DUAL CORE "i3"CALLED A i5 THAT IS STILL A DUAL CORE.
-LOOK UP GAMING BENCHMARKS FOR THE GAMES THEY PLAN ON PLAYING FIRST. Some will run fine on the skylake celeron. others need a quad core and run better on the FX than the budget celeron. The skylake i3 is too expensive so the FX-8320E is a better deal. Try to get the 8 core FX instead.
-The FX needs a decent cooler. If all of the games support dual cores, the skylake or Kabylake celeron is worth considering. That leaves a easy cpu swap to a i5 later on. Try to get a board that supports Kabylake i5's.
For about $100 you should be able to find a mini-tower case like a HP-Compaq or E-machines with a 1156 or 1155 i3 or pentium dual core cpu, or the 3.0ghz core2 cpu or Athlon64 X2,X3 or X4 windows 7. A complete working system. You will need a new gaming power supply and maybe a case fan or two. But that should work ok with the not so fast GTX650ti. MAKE SURE IT COMES WITH WINDOWS 7 x64 OR HIGHER, AND YOU SEE IT WORKING AS IT IS USED AND KINDA OLD. Bring the card with you and see if it fits in the case and slot.