kinda lazy to build something, butw i would approach the budget is to utilize a pentium g series cpu(pentium g2120 preferably) with a budget b75 motherboard. get the rest of the necessities and pump it all into a gpu afterwords.
would look sometng like
p g2120 95$
b75 mobo ~75$
ram ~35$
case + psu ~100$
hdd ~70$
rest would be allocated into gpu. which pricing wise, will probably be a 7770 for that relative price range