From the PHB: "Alchemist's fire is a sticky, adhesive substance that ignites when exposed to air."
Question #1: If alchemist's fire burns upon contact with air, how does an alchemist get it into the flask?
Question #2: Given a typical barrel, how is anyone going to fill it with alchemist's fire in the first place.
Ignoring question #1 for now, I have a solution to the problem posed by question #2. Instead of filling a barrel with alchemist's fire, fill it with flasks of alchemist's fire.
Oh, and although the "launch this into the castle" idea is plenty of fun, here's another one. In a dungeon with lots of curvy, hilly paths, put one of these on the top of a hill where a band of adventurers might be coming up sometime soon. Either have a trap set to loose the barrel, or have a guard sitting there waiting to move the chock out of the way.
As the barrel rolls down hill, the bottles inside are clanging around. Sooner or later some of them break, catching the barrel on fire. As more break, a nice layer of burning, sticky liquid coats the hill. Thanks to gravity, the spilling liquid ought to cover a nice, big parabolic area on the hill (go with a cone shape to fit the rules), and, with any luck, the party will be half-way up the hill by the time all the real damage is done.
Now they have to go either the rest of the way up, or back down, but no matter what, the floor is burning with a sticky, gooey mess. Maybe 2d6 fire damage every round, if they move at half speed, 4d6 if they move at full speed? Either way, they can't put it out until they get out of the fiery pool at their feet.
Oh, and if someone is hit directly by the barrel (treat this part as a trap rolling to hit?), then that person takes maybe 1d6 from the barrel and an additional 5d6 fire (and, of course, this should be a x3 weapon on a critical).
Dave