To be honest, not even touching the CPU/motherboard compatibility issue, the i3-6300 would be so-so. Ideally, you'd need a minimum of an i5-6500. If you read the article on cores vs games which TomsHardware just put out the other day, the whole gist of it is that previously, games were not highly threaded. This is NOT the case any more. They put an i3-6300 in the tests along with 4/6/8/10 core chips, disabled hyperthreading, and tested them. For the most part, with a high end GPU, the i3 (in effect, down to a dual core) got smoked. Hyperthreading helped a fair bit, but the minimum is now up to a quad-core CPU as more and more games and engines are highly threaded. So - the i5 will do the job pretty well (better than the i3 even though it has hyperthreading to simulate 4 cores).
On to your upgrade. What you have in mind won't work, different CPU lines utilize different sockets for the most part. There are some which would work (such as Sandy Bridge and Ivy Bridge, and Haswell and Broadwell) but you're talking about what is probably an LGA775 CPU (Core Duo) vs LGA1151 (SkyLake). They are just NOT compatible. At all.
So you would need to upgrade the CPU, the RAM and the Motherboard. And this isn't even covering what you're using for a GPU.
You have two options really, either of which would be a huge jump in capabilities:
Haswell build:
i5-4570 Haswell LGA 1150 CPU. Compatible H81 based board of your choice. 2x4gb or 2x8gb of DDR3 RAM. This would probably cost somewhere in the $300 range ($175 for the CPU, $50 for the board, $50ish for the RAM + tax). It would be a perfectly good base for a build. Depending on the RAM in your old machine, if it's DDR3 theoretically you could reuse it and save $50.
Skylake Build:
i5-6500 CPU, H110 based motherboard of your choice, 2x4gb or 2x8gb of DDR4 RAM. Would cost a little more, but a little more future proof too. Figure about $325-50. CPU $230, motherboard $55, RAM $65ish).