I am looking to make a PC with AMD FX-8350. It's 125 watts, so will it be cool enough to play and record Minecraft? Or should I get a CPU cooler, if so which one.
You shouldn't need an aftermarket unless you're overclocking. Under load it shouldn't get hotter than the higher 50s, but if you want an aftermarket cooler in case you want to overclock or if you're just not comfortable with the temps you're receiving.
If you are not planning on overclocking, the stock CPU should be able to handle the load you are putting onto your CPU. You mainly only need an aftermarket cooler when you are overclocking. However, an aftermarket CPU cooler will ALWAYS keep your CPU cooler than the stock one and will extend the life of your CPU, so laying down the extra money now may save you money in the future even if you don't necessarily "need" it.