You only need to get the final blow with the broom for it to count, I would recommend enchanting your main weapon to the max and un-equipping any pets as they can get the final blow instead.
Upgrading the broom only increases speed so its up to you if its really worth it or not but you'll still probably only need it for the last few hits
As far as my experience, there's no way to track if you were successful or not until the achievement pops on the last boss. Just be sure to restart if you think you miss the broom kill