The discovery says, "The effects of the smoke created by an alchemist’s bomb duplicates the effects of cloudkill instead of fog cloud [that's created by the prerequisite discovery smoke bomb], filling an area equal to twice the bomb’s splash radius for 1 round per level." However, what that means isn't entirely clear.
Maybe it's precisely like cloudkill…
This reading emphasizes the bomb's effect's similarity to the spell cloudkill. Using such a reading, the cloudkill effect behaves as any other cloudkill effect, so it "moves away from [the affected area and, probably, the alchemist] at 10 feet per round, rolling along the surface of the ground"; the GM must "[f]igure out the cloud’s new spread each round based on its new point of origin, which is 10 feet farther away from the [original] point of origin where you cast the spell"; and the effect "sink[s] to the lowest level of the land, even pouring down den or sinkhole openings[, but i]t cannot penetrate liquids, nor can it be cast underwater."
This GM thinks—and this fine answer ultimately agrees—that this reading likely overextends the mandate of the discovery poison bomb. That is, this isn't a particularly superpowerful reading or anything—the level 12 alchemist, at this point, has devoted 1/3 of his discoveries to smoke bombs that in a much smaller area and for a much shorter duration create an effect that a wizard's been creating for the last three levels—, but, simply, one that, in this GM's opinion, goes beyond the text… and, not inconsequentially, is also enormously complicated, making this GM dread the alchemist ever hucking even a lone poison bomb.
…Or only the smoke has an effect like cloudkill
This reading emphasizes the opening description of the discovery poison bomb: "The effects of the smoke created by an alchemist’s bomb duplicates the effects of cloudkill." So the smoke created by the discovery poison bomb is unlike fog cloud's effect that
obscures all sight, including darkvision, beyond 5 feet. A creature within 5 feet has concealment (attacks have a 20% miss chance). Creatures farther away have total concealment (50% miss chance, and the attacker can’t use sight to locate the target).
Instead, the effects of smoke created by the discovery poison bomb are like the effects of the smoke created by the cloudkill spell, which are
similar to a fog cloud, except that its vapors are yellowish green and poisonous [so still having the sight-blocking effects above]. [Plus t]hese vapors automatically kill any living creature with 3 or fewer HD (no save). A living creature with 4 to 6 HD is slain unless it succeeds on a Fortitude save (in which case it takes 1d4 points of Constitution damage on your turn each round while in the cloud).
A living creature with 6 or more HD takes 1d4 points of Constitution damage on your turn each round while in the cloud (a successful Fortitude save halves this damage). Holding one’s breath doesn’t help, but creatures immune to poison are unaffected by the spell.
Thus, under this reading, the effects of the smoke—the damage it deals and its appearance—are like cloudkill, but the GM has ruled that movement of the smoke away from the creator and down into pits or warrens or whatever is not an effect of the smoke so the poison bomb smoke remains stationary as per the fog cloud spell.
While this reading is a little strained, it's far more playable than the first alternative. Considering the sheer number of poison bombs an alchemist at this level can drop in only a few rounds, keeping track of the number of drifting cloudkill effects—even for a few rounds and especially in a highly-populated area!—is a GM nightmare, making this, too, a reasonable ruling.
Note: While the question's title asks How does a cloudkill generated by the alchemist discovery poison bomb naturally disperse? that answer is as per the fog cloud spell: "A moderate wind (11+ mph) disperses the [effect] in 4 rounds; a strong wind (21+ mph) disperses the [effect] in 1 round," or, alternatively, as per the discovery smoke bomb, the effect dissipates naturally in 1 round/alchemist level. However, I think the actual question is Does a poison bomb's smoke move like the smoke from the cloudkill spell? but the answer to the title's question is here for completeness.