Upgrading CPU and Possibly Mobo

[Cathbadh]

Last year I bought my son a cheap refurbished Dell for him to start gaming on. It runs Minecraft and a few other games, even Heroes of the Storm pretty fairly. However he's looking to get into more modern games including Overwatch. It has been a long time since I upgraded a PC over buying one, and am a bit lost.

Currently he's on a refurbished Optiplex 755 running a Core 2 Duo at 2.66GHz. I unfortunately don't have any info on what motherboard the chip's attached to, so I don't know if I can upgrade to a better chip or if I'm going to have to replace both the chip and the motherboard. He also is running a pretty crappy ATI Radeon HD 5450 for graphics which I plan on replacing with a GTX 750ti.

The minimum requirements for Overwatch are a Intel® Core™ i3 or AMD Phenom™ X3 8650, although I'd like to get him something a little better than minimum requirements so he is able to play new releases in the next year or two, even if they're at low settings.

My budget is low, around $200-$250 USD for the processor and (assuming I need a newer one) motherboard.

[Fascinate]

Gonna need memory as well:
PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i3-6100 3.7GHz Dual-Core Processor ($110.99 @ SuperBiiz)
Motherboard: Gigabyte GA-H110M-A Micro ATX LGA1151 Motherboard ($45.99 @ Newegg)
Memory: GeIL EVO POTENZA 8GB (2 x 4GB) DDR4-2133 Memory ($32.98 @ Newegg)
Case: Fractal Design Core 1100 MicroATX Mini Tower Case ($28.99 @ Newegg)
Power Supply: EVGA 450W 80+ Bronze Certified ATX Power Supply ($37.49 @ SuperBiiz)
Total: $256.44
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-08-15 01:26 EDT-0400

That said, old cases sometimes do not have standard mounts for motherboard or PSU. If you can push the max of your budget it will really be worth it to get the case and PSU to avoid any compatibility issues and have peace of mind with quality power running to the components.