This challenge is very easy, can be achieved in only a few minutes and you don't actually need to own The Dude weapon to do it.
Simply bring up the menu and go to the challenge section under log, scroll down and travel to Weapon Challenge: The Dude, once loaded press
to start and redo until the achievement unlocks, which for me was halfway through my second attempt.
If the Weapon Challenge: The Dude is not yet available, you'll need to further play through the campaign missions until The Dude can be purchased from Two Hat Jack, at which point I believe is also when the challenge becomes available. However if not just buy and equip The Dude, then redo any Night Defence challenge instead, using it at the barriers and it shouldn't take long before the achievement unlocks.
P.S. Doing the weapon challenges over of weapons you actually own is a quick way to level them up and any amps you have equipped to them will also work during so can help to easily obtain a higher score (I personally found the Twist of Fate amp best for this).