This is actually much easier than I thought. All you need to do it to win some money (so basically one round of each of the mini games). I thought this was popped after completing all the mini game achievements but it is in fact just for winning one round of each of the games. The games are:
OK this minigame is effectively part of the main quest line, you will find this in the Hagar settlement at the start of the game. Talk to Loosum Hagar and basically complete the Wingstick tutorial
-Rage Frenzy (Card game)
This game is first unlocked in the Wellspring Town Bar. Speak to Hendrik to start a game. Just try to beat his hand on easy. If you do not meet Hendrik do not fear, you can also do this when you reach 'Subway Town' by speaking to Teague.
This is the illuminated board game found on the floor near the outfitters in Wellspring. All you need to do is win one game, it's pretty straight forward. If you come away with more cash than you started then you've won! It's a simple dice game.
-Five Finger Fillet
This can be played in both Wellspring and Subway Town bars by speaking to Abbott. This game involves you playing chicken with your virtual fingers and a knife! Not recommended IRL! Complete the first round and then simply take the money rather than the option 'let it ride'.
In Subway Town, after coming out of the Garage head to the furthest staircase and head upwards. At the top of the stairs is a dude who looks like a pikey boxer with a banjo. Select him and play the most annoying game of guitar hero ever devised for one round, then back out taking the money.
This should net you the minigamer achievement, no sweat!