When Mythbusters tested this one, I got the impression that it just wasn’t worth it. Supposedly, you have to burn through a filter for each filtration. The more you filter, the closer it gets to tasting like top shelf vodka (and the cheaper the vodka, the more filtrations it needs), but according to a vodka tasting expert in blind taste test, the top shelf stuff is still better than stuff that’s been filtered 6 times. Considering a bottle of top shelf vodka will cost you $25-35, a similarly sized cheap bottle will run $10-15, and Brita filters run $4-6 each, eventually, it really becomes more worth your time and money to buy the nicer stuff. Four filtrations will get you to a similar price point. Going up to 6 filtrations is a colossal waste of money (not to mention time). You could buy better than top shelf for that money, which would supposedly still taste better than filtered.

Tl;dr: if you really want good vodka, it’s probably more worth your time and money to buy the better stuff from the start. If you’re planning to mix with it anyway, save the Brita filters since you won’t be able to taste it much over the mixers.