A FX-6300 has a passmark rating of 6344 and a single thread rating of 1408.
You might plan on being able to overclock the stock 3.5 to 4.5 or so, a 28% oc, depending on your chip.
That would theoretically give you a new rating of 8120 and a single thread rating of 1802.
That is excellent if you have the multitasking apps to keep all threads mostly occupied.
OTOH, if you will be playing games, the single thread rating is more important. Few games can use more than 2-3 threads.
FX-6300 will be $85, a 970-ud3p motherboard will be $85, and a top cooler like a H1001 will be $100 for a total of about $270.
Not bad for a $270 budget.
If your use is for gaming, then a $125 Intel I3-6100 and a $50 LGA1151 motherboard will have a passmark number of 5500/2102. Great for gaming, but not as good for multitasking. it is, however cheaper and better if gaming is your main thing.
I do not think the non K i5 processors are nearly as good a buy as the I5-6600K and a z170 motherboard.
If you live near a microcenter, you can buy a i5-6600K for $180 and get a discount on a z170 motherboard perhaps$80
add a $30 cooler and you are at $290. some $20 more than your I3-6300 budget.
What do you get??
I5-6600K @3.5 has a passmark rating of 7791 and a single thread rating of 2126.
some 87% of samples will oc to 4.6. a 31% oc.
Apply that to the stock numbers, and in theory you should have a cpu with a 10206 rating and a single thread rating of 2785.
Not bad for $20 extra.