No, it's actually way easier than that.
Got a stand mixer? Or a hand mixer? Good. Put the cream in a (metal) bowl, and go to it with the mixer. First it'll turn into whipped cream (mmmm) then it'll be sort of grainy whipped cream, then it'll be butter and buttermilk, at which point it'll go all sloshy so you should shut off the mixer before you make a mess.
Dump it all in a collander or sieve in a bowl, lined with cheesecloth or a loosely-woven dish towel. The liquid will drain through to the bowl and you'll be left with butter in the cloth. Gather it up in a ball, and tighten the cloth so as to squeeze out as much liquid as possible, but stop if the butter starts oozing out of the cloth.
Run your cold tap for a little while to make sure it's as cold as possible (or use some ice here) and place the butter in a deep bowl filled with the cold water. Squidge the butter with your hands, squeezing out all the remaining buttermilk and pressing it back into a ball as you go. Change the water frequently. When it doesn't go cloudy anymore after a few squidges, you're done.
This part is optional: Rinse your mixer off and put the butter back in it. Sprinkle some salt on and turn the mixer on just long enough to mix the salt in well. Then use a spatula to scoop out the butter (mixer-licking also optional) into a nice little bowl, cover and refrigerate. Homemade butter is softer than store butter so you can keep it in the fridge and it'll still be spreadable, mostly.