HITMAN

The description should be "beat the high score on a contract"

Update : It sounds like my explanation was a bit weird/not complete, please find some clarification below (It sounds that you have to play it 3 times instead of 2) :

It could be done on your own contract by beating your best score. So the trick is :
1st playtrough : You need to create the Contract,
2nd playtrough : Play it slowly to score a low score
3rd playtrough : play it faster to get a better score and pop up the achievement.

I have done it on the first map with the jet fighter :
To create the contract :
1 - Start the contract creation,
2 - Mark up the 1st guard as target
3 - Wait 20 secs
4 - Kill the guard with gun
5 - Escape using the button.
Replay the contract and set a low score :
1 - Start the contract
2 - Kill the guard with gun as soon as you spawn
3 - Wait 10 secs
4 - Escape using the button.

If you do not pop the achievement you havee to replay a 3rd time as below :
Replay the contract and set a high score :
1 - Start the contract
2 - Kill the guard with gun as soon as you spawn
3 - Escape using the button.

As you should be faster, you get higher time bonus therefore you beat your highest/previous score and pop the achievement.

Remark from Megamonkeymania : Replay it as soon as you do your first run to avoid other player to score the high score

heres a nice simple video guide for anyone thats stuck or confused :)

To get this achievement, you have to beat the high score on a created contract. I guess the "proper" way to do this is from the Main Menu, select contracts using RB, then select a contract and play that one, with a view to getting the high score.

However, the easier (boosting) way to do this is to create your own, play it to set a score and then beat it.

The knack here I found, and different to Polak's is that if you make it too easy, before you get chance to replay it, someone could have beaten it! When I first did this, with the objective to kill the first guard, within 3 mins, 2 other people had played my contract!

What I recommend you do is therefore create a contract that has a bit of difficulty within (but not too hard) - for example, in the palace, I set it as using a melee weapon on a random, who was in the middle of the palace, but obviously I knew the quick way to the weapon and then the target. This way you can play it the first time as fast as possible, but leave a pause before you end the mission (to set a points target), and then replay it, but slightly faster, so as to get a higher score, and as such get the achievement for beating the high score.

By doing it this way, this ensures that if someone loads up your contract in between you doing it, for a couple of minutes they are not going to be 100% sure of the fastest/quickest route to complete the contract, and this should give you enough time to make sure you can beat your own contract. Good luck!