You first need to get Sherpa (a npc that fixes the elevators) to show up to start this. After your first flawless boss fight, he will show up outside of the second floor's elevator. Edit: I now believe that you don't need to flawless to get him to appear the first time; he may just randomly show up outside of the 2nd floor elevator. Note that even if you get hit during the boss fight with armor on, it will not count as a flawless victory. You cannot get hit at all!
Once you talk to Sherpa on the second floor, he will go back to the Breach. Go talk to him there once you are back and he will note that the elevators are broken and he will fix them. Enter the Gungeon and reach the second floor again. Once the elevator drops, go ahead and follow down into the elevator shaft. You will enter into a room with Sherpa and some electrical wires. He will need some items from you in order to finish his repair. These items are:
1) Three blanks
2) Three keys and 120 Coins (Note that you need both of these at the same time)
3) 10 Hegemony Credits (Those green "H" icons you get when defeating bosses)
4) The Master Round 1 which you will get when you get a flawless victory at the first floor boss
Note that these must be given to him in order except the Master Round 1! So if you flawless the boss and get the Master Round 1, you want to still talk to Sherpa and give him the item even if you haven't given the 3 keys and 120 coins for example. Not a bad idea to get that item out of the way if you can!
Everyone has his/her own play style, but I found myself performing better with the Marine. The Hunter was a close second for her one hit crossbow on the first floor. Do not worry too much about this achievement until you start unlocking more powerful guns from the shop Ox and Cadence run. For the second step, if you ever see keys for sale during your run buy them! Keys are much harder to come by than saving up for the 120 coins. You can always defeat the second floor boss in order to get enough coins for his requirement.