The Orange Box

While going for the Hardcore achievement, you will get this fairly easely. But you can also get a friend to kill you five times and then kill him. do that five times and you have it. Or play really bad in a normal game then you will get an nemesis and then go after him to kill him.

This is very easily boosted in a 1v1 with another friend. In fact, you can do this, Hard to Kill, and Grey Matter all in the same session. This solution assumes you want to do all three at the same time.

Set up a match with all private slots, and invite your friend in. Set the timer to unlimited.

P1(red) sets up as sniper and goes to blu base. Headshot P2. Stay near the spawn spot, and have P2 come out and get his headshot. Do 5 straight headshots (Hard to Kill ach pops at this point).

Before you get the 5th headshot, you'll see that you're dominating P2. After the 5th headshot, switch roles, and let P2 headshot you 5 times, and have P2 move to P1 base. P2 gets a revenge kill for their first kill of you. Let them go forward to get 5 in a row (if they need Hard to Kill ach), and then go back and forth, with each player taking 4 headshots in a row at a time.

By doing it this way, you knock out both the Nemesis ach and the Grey Matter ach. All in, I think we took 30-45 mins tops to knock this out. It's really rather quick to get all of these.

This one is fairly simple. You’re bound to get dominated by other players sooner or later. Once you are being dominated just kill the person with a boxing glove icon above their head. Do this 5 times and you will earn it.

Here is a very good description of how this works from the Team Fortress 2 wiki:

"A Domination is a reward received for killing, or assisting in killing, one particular enemy four times in a row. Accompanied by a distinct sound cue, it adds a special icon to the scoreboard. If you Dominate a player, you become their Nemesis, and this is indicated to them by a small icon appearing above your head. If your Nemesis kills you, you will lose your domination and status as their nemesis – this is known as a Revenge kill. "

Source: http://wiki.teamfortress.com/wiki/Domination

So you need to be killed four times to be able to get a revenge kill by killing your nemesis, not only three as one of the solutions suggested.

A good way to get this fast would be to boost it with another friend in a closed game session (8 players max and 8 private slots) by staying close to their spawn point and keep killing them, maybe even use a turret for that matter (close to their spawn point).

This is for all the new people that does not know what it is. In games, to get a Nemesis, you need to get killed 3 times in a row by the same person. Once you kill him back that counts as a revenge kill, get 5 of those on anyone. Bloop, achievement unlocked.