The easiest possible way to complete this achievement and the other Holodome achievements is to be a high enough level and to be playing on Normal mode. If you have a group of friends then you probably would be good if everyone was around level 35 or higher. If not just wait a little bit until you're a higher rank. You should have no trouble finishing up the Holodome at level 45+ with decent weapons and equipment. You can accept the Holodome quests from the quest menu and travel there from a fast travel. After completing a round, you must head back up to the claptrap that gave you the mission and turn it in. Then accept the next mission. :)
I would also like to point out one last thing, there is ammo in the crates but it runs out fast if you are not buying ammo before/after every round. So buying ammo in between rounds is a good idea, if you are afraid you're going to run out of ammo, you can always fast travel somewhere and come back after each round to reset the crates. If you do however happen to run out of all ammo in the area, you can either pick up or switch to a different weapon you haven't used. For example: You run out of all Shotgun, Sniper and Assault Rifle ammo, you can switch to Rockets, or SMG ammo.
The videos are to help you if you're stuck, the person playing is level 50, so only watch if you aren't sure on something or need help. Good luck
If you have any questions or concerns, please leave a comment and I will do my best to try and help. Round 1
Simple round it has only 3 or 4 waves and if you're a high enough level, you should have no problem. You can also complete the bonus objectives if you need to work on your badass rank or challenge.Round 2
Basically the same as round 1 but with stronger enemies. It should only have 3 or 4 waves with another bonus objective.Round 3
Keep an eye out for the spinning blade in the middle, if you fall in there it will kill you instantly and when you die, you will have to restart the entire round that you are on. If the blades don't show up on Round 3, they will show up on Round 4 or 5 so keep your eyes open. There will be more enemies then round 2 and they start to get even stronger. They also add in an extra wave you must complete. Don't worry though it's not too bad :)Round 4
Now you will notice that rounds start becoming longer as there are more waves to complete. Round 4 is where it will start getting somewhat challenging. Don't let that scare you though because it's pretty easy if you are playing as a higher level character. Remember stay away from the blades and always watch where you are walking. Don't back up because you might fall and hit the blades.Round 5
You're going to have to start fighting stronger enemies now and there are 6 waves to complete. It will take a while if you are playing solo but just take your time and you will get it. The best advice for the last wave is to stay on the lower grounds and stay back. Fight off the enemies in the area and run if the enemies start over running you. Once the place is cleared, go to another area of the map and fight off the enemies. I usually froze the Robot enemies and then switched weapons to whatever else and went for critical hits. Once you start to stop seeing enemies, search around because there are usually a few that stay in the area they spawned at. Take your time and be careful and you should unlock the achievement. Don't be mad if you die, just come back to it later when you feel you are ready to give it another try.
You don't need to complete the Badass round so the achievement should unlock once you turn in the mission to the claptrap at the top. Also hosting your game might also be a good idea because I heard of people not unlocking achievements after turning in quests if they were not the host.
Credit to Thabuki on Youtube for the videos