The key facts to remember are to be Human and to choose the Commoner story line. Everything after that doesn't matter much.
Class Choice: I have tried most variants of all classes and learned that a short bow ranger is the quickest for me to get through the story with. I can get a key in ~20 minutes with this build.
I also had some success with long bow warrior and staff guardian. Staff ele is also decent but you can die quickly if you aren't good.
Gear: This will only matter to those who bought the special edition of the game. You get a hero's band ring and a golem banker to start with on all new characters, which is perfect for key farming. You get the additional +2 to all stats and the ability to open up your bank to access gear and consumables.
For my short bow ranger build I only bought the level 1 coat, pants, and boots. You can also buy shoulders and a head gear if you want to kill faster. Additionally I bought a condition damage short bow with minor sigils of agony and fire. Condition damage is very strong at the beginning of the game. You can also use power stats if you'd like but power is greatly reduced when you get into the level 6-10 story missions and start hitting glancing blows on a lot of your shots.
Walkthrough:
To start off with you will be placed in Shaemoor. Do not talk to the green asterisks like you are supposed to. Instead follow the arrows on my map below. Run down the hill, through the watermelon fields, down into the river, and up the hill. If the giant hands haven't spawned then you will need to run across the bridge into the garrison to spawn the champion centaur.
After killing the starter boss you will be placed in a house in Shaemoor. Immediately exit the cut scene and accept your reward. Run out the door and talk to the map guide person. Immediately exit out and follow the giant green arrow on your map across the bridge. Be sure to get the first waypoint across that bridge unlocked then head to the farm on the right where you will feed cows, water plants, and kill wurms to get your first heart done.
Upon completion of the heart read and delete the mail you got. Then waypoint to your home instance in Divinity's Reach. Run straight to the bar to talk to your friend and exit the cut scene immediately. Kill all of Big Nosed Ted's henchmen downstairs. Utilize your Stalker pet's F2 skill for 5 stacks of might whenever possible. You will quickly kill everyone with this much might. When you head up stairs following after Ted and Jake, be sure to shoot Ted while he's still yellow. This will skip his little cut scene and also allow him and Jake to damage each other. Once all the bandits are taken care of head down stairs and get your reward. You should then take the power gloves for your reward and equip them.
Your next mission is desperate medicine. Waypoint back out to Queensdale next to the Lieutenant and talk to him. Then run to the farm and pickup some hay on your way to the bandit caves. This helps you get swiftness every few seconds on your way to the caves. Once in the caves again utilize your pet's F2 skill for might whenever possible. Kill the first 2 guards and run straight to the Moa's. Kick the King Moa's butt and watch them run. Wait a couple seconds then follow them as most the bandits in your way should follow them as well. Head straight for the ramp on your right in the next room and go up. You can one shot the few bandit recruits who spawn, then take care of Jake and his body guards. Grab your medicine from the supplies and finish off Jake. Waypoint back to the Lieutenant to tell him your news then waypoint back to your home instance in Divinity's Reach.
Now you will be on the undercover mission. Waypoint back to Shaemoor and run along the cliff's, through the cemetery, and to the bandit hideout. Once in you will have to take down 1 guard and take his clothes. Then enter the hideout. Once inside run straight and drop down next to Johnny. Talk to him first then run across the water and kill the drakes. Take the fuses from the box and go to Blaize. After talking to Blaze you can then kill the final rifle wielding woman to end this mission. Head back to your home instance waypoint in Divinity's and go to Logan's office. Make sure to grab the waypoint by Logan's office on your way in. Once you talk to Logan be sure to take the Orphanage!
Now head back to your home instance and go to the orphanage which is on your right as you enter the mission. Do not take down the door. Instead target the 4 bandits directly in front of you inside the orphanage behind the wagon. You can shoot through the walls and take them down 1 at a time. Utilize your pet's F2 skill for your stacks of might. This is the longest part of all missions as Blaize clears conditions and has a lot of health and you will be a few levels below them. Once you kill these 4 bandits then you can take down the wall and take out the next group of 4. Remember you can shoot through walls so be sure to take advantage of that to avoid taking any damage. After you kill them you can head upstairs and finish off Ted. Then run straight to the hospital on the other side of the instance so you can talk to the Lieutenant as soon as the cut scene pops up.
The final part of the story missions will be heading back to Logan's office. Talk to him and the countess then head downstairs into the far jail cell. You should hide behind the bars for a while and pop out and shoot every once in a while to get tags for loot. You basically do no damage and take tons of damage at this point so staying alive is "key" here. The sword illusions will do all the damage you need. Once finished you'll get your key. You can use your golem banker to deposit your key and gear and be on your way. If you don't have a golem then you will need to press the swords button at the top of your screen and enter the mists. Once in the mists you can run to the Lion's Arch portal straight ahead and access your bank in LA to deposit the key.
After you have your key you can delete your character and start over. Like I said it's about 3 keys per hour for me. The point of the keys is to unlock black lion chests which will yield tonics, boosters, hair and makeup kits, dyes, mini pets, and black lion claim tickets.