You will need 3 friends to do this. Set up a private match with the Score Limit set to 500, that is important.
Normal cards are worth their number, special cards like shift, skip, +2 are worth 20 points and the two black card types are worth 50 points each. When a round is over, the winner gains points according to the cards, the other players have still on their hands. For example: 5 + shift + black card = 75 points.
Now there will be two steps.
In the first round you will have to try to get a high score but not over 500, if your score is over 500 at the end of the round you have to try again. Your friends simply draw cards while you try to get rid of your own cards. Your friends should always check the points on their hands, so you won't get over 500 in the first round.
In the second round you simply have to make sure your friends have enough points in their hands to push you over the remaining points to 700, before you drop your last card. For example: You scored 450 points in round one, so you have to score at least 250 points in round two.
After the results screen, it should pop.
Edit: NitrousSpeed pointed out in the comments, that you can also do this in only one round, by everyone except yourself having at least 50-60 cards on hand. Just do what you like better :)
Edit 2: Good advice from gamer evil terrorr: If everyone in your group needs this achievement, do a 2vs2 game, this way you are twice as fast. Furthermore enable the rule "draw until match" to speed up every round.