Is there a way to play as another faction (i.e. Bandit) on Skyrim for PC?

Alright, first you need to pick a starting location. Go here: www.uesp.net choose a location there, and copy the code to the location. This'll be your starting area. Eg, Valthume. Good alternate start as an adventurer who wandered in there for treasure.

Now open Skyrim.ini (NOT SKYRIMPREFS!!!), this is usually here -

C:\Users\%USERNAME%\Documents\My Games\Skyrim

Under the [General] tab, type in

SStartingCell=(area ID)

and save it. Eg, SStartingCell=Valthume00

This will put you in Vathume in a new game the next time you launch Skyrim.

Once saved and closed, head to the folder Skyrim is located in. Create a text document there and call it bandit (not bandit.ini not Bandit, it has to be bandit). Open it and type in the following:

player.removeallitems
;Leather Armor, Bracers and Boots
player.additem 3619e 1
player.equipitem 3619e
player.additem 13921 1
player.equipitem 13921
player.additem 13920 1
player.equipitem 13920
;Steel Sword and Shield
player.additem 13989 1
player.equipitem 13989
player.additem 13955 1
player.equipitem 13955
;Hunting Bow and Iron Arrows
player.additem 13985 1
player.additem 1397D 25
;Gold and Lockpicks
player.additem f 100
player.additem a 15
;Bandit Ally Faction
player.addtofaction e0cd9 0
;Create Character
showracemenu

Save it, and exit.

And there you go, just launch up Skyrim and begin a new game. You'll then start in the area you wanted. Then, type

bat bandit

This will give you Bandit starting items, a little gold, align you with the bandit faction, and give the starting character menu to customize your race, name, etc.

Some things to note:

As long as the alternate start options are setup, if you start a new game it will skip the first cinematic and you will not be forced to start the main quest-line.

Sometimes the game will auto launch a new game for people. I haven't found a fix for this yet.

This might crash some games and not work. That means the starting area code is wrong. Fix it, and launch it again.

Usually, as soon as the game is started, the character creator will not appear. So use the console command openracemenu and hit enter, this'll allow you to customize your character to your liking.

If you don't appear in the alternate start you created, you probably edited the Skyrimprefs.ini for the starting area code. It has to be under Skyrim.ini, not prefs. Be sure to change that.

Any alternate start mods will prevent this proccess from working. Turn off ALL alternate start mods and you should start in your custom start.

To start the main questline, go into the console and type in the code for the MQ quest you want to begin in. www.uesp.net will help you find these codes.

You CAN explore Helgen before it was destroyed by Alduin, but there are no quests/interiors avalible.

There's a (free) mod called Live Another Life which gives you a number of starting options - one of which is to start as a bandit. This sounds like it'll be exactly what you are looking for.

From the mod's readme:

Outlaw in the Wilds

You will begin as a member of a group of bandits. For the sake of variety, the group of bandits will be chosen at random. You will begin with standard bandit level equipment. The bandit faction at the start location will remain friendly to you as long as you don't piss them off first. Tread carefully though, you WILL have a bounty of 500 gold in the hold the game chooses. The guards will not take kindly to seeing you trotting into a civilized area unless you're prepared to pay up.

You will require Skyrim patched to 1.9.32.0.8 but I don't think you need any DLC.