Culture fresh buttermilk with yogurt
You seem to have gotten the process backwards. You don't culture the buttermilk. You culture the milk, then whip the butter, and the rest is cultured buttermilk, at least if you are going for traditional buttermilk. If you want modern buttermilk, you don't whip anything, just culture the whole milk and consume the result. 

The original method would include "self culturing", so you just leave your freshly milked milk at room temperature until it is done (soured). This is not considered acceptable by modern safety standards, even if you have your own cow. If you are starting with milk you bought somewhere, it is not even likely to work. So your best bet to get buttermilk culture is either a package of storebought buttermilk with live cultures, or to look if some specialty store sells the culture itself, possibly online. 

I cannot comment on the perfect temperature/time conditions for buttermilk, they are also likely to differ between strains just like they differ between yogurt strains. Maybe somebody else can post the usual ones in another answer.