How do I do a rampage?

Solution 1:

If a champion manages to get multiple kills without dying, they go on a killing spree, which increases the amount of gold you get for killing them. The announcement for the killing spree changes based on how many kills in a row they've achieved, going from "Champion is on a killing spree!" at three kills up to "Champion is legendary!" at eight or more kills.

To answer your specific question, a rampage is announced when a champion kills four champions without dying.

For the full list of killing spree announcement names and bounty rewards, visit LoLWiki's page on Kills.