Same deal here: murder everything. There’s a twist, though, because now save points will display how many monsters you still need to kill, in prominent red. If this number disappears you have moved off of the Genocide Route! Be careful not to save if this happens, because you’ve apparently done something very wrong! Linger in the middle sections of Snowdin, before you reach the town, and kill everything before moving on too far. (Try not to fight Jerry. He takes too long to kill physically.)
In addition to the random encounters, you’ll want to kill Doggo, both of the Dogi, the Greater Dog, and Papyrus. Do all of that and you’ll continue along this bloody path.