Is it possible to make a computer within the price range I provided for (medium) gaming (minecraft, league of legends and other games like that), streaming and uploading videos to YouTube.
Yes, of course this all depends on where you live, where you are willing to shop and how quickly you need the parts. If you live in the United States, live near a microcenter or are willing to wait for sales, it is very possible.
I am assuming for this computer you would need to purchase an operating system. Do you also need peripherals? (mouse, keyboard, monitor) and if you do need those things, will they be included in the budget?