Multiplayer: You have to level up to 65 to get the Flamethrower. Once you get to 65 you'll be able to use it in perk slot one.
Campaign: I think you can only get it in the missions, Burn 'em out and blowtorch and corkscrew, It's given to you at the start, or vendetta, through a glitch.
Nazi Zombies: You can get it out of the random weapon box, which costs 950 points, it's the wooden box with the glowing question marks on it, located in the help room on Nacht der Untoten, the power switch room on verruckt, the room just below the starting room on Shi No Numa and the power generator room on der Riese. On der Riese, when Pack-a-Punched, it becomes the FTW nitrogen cooled, which has a longer overheat time, a faster cooling speed and higher damage.
p.s. In multiplayer if you have the flamethrower you can be easily killed by somebody shooting the fuel tanks on your back, which WILL incinerate you.
p.p.s. You can use the flamethrower against some foiliage in multiplayer maps and nacht der untoten to burn the foliage down.
p.p.p.s. If you use the flamethrower in multiplayer you don't get a ghillie suit with a bolt-action or PTRS-41 rifle.
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In single player mode it will be either given to you at the start of a mission, or will be provided at a certain point.
In multiplayer it is a Tier 1 perk which is unlocked at level 65.