Might as well give a why not answer to this...I'm not sure on feasibility of alot of these ideas, but the Greeks were amazing engineers in their time and might be able to get away with it. You would not see guns by any means, but I'm pretty sure if the Greek people had knowledge and access, they'd come up with some pretty creative uses.
Hand thrown bombs. It actually takes pretty simplistic technology to build a bronze sphere around the size of a baseball (doesn't need to be a perfect sphere either) with a small hole. The bomb (grenade?) is then filled with brass shards and saltpeter. Fuse it lit, bomb is thrown, brass shards everywhere. Extra points it the soldier doing the bomb throwing is mounted and exceedingly mobile (skip horse archers, lets go with horse bombers. Horses would need some pretty intensive training not to spook though)
This could readily be scaled up to a larger catapult thrown bomb. Same theory as above, load a larger bronze sphere with saltpeter, bronze shards, and something to provide a spark upon impact (flint and steel style). You now have a bomb launching catapult...probably comes with some dangers and unintentional detonation may cause friendly casualties.
It's important to note saltpeter is not gunpowder...the gunpowder we use came significantly later. This would function and behave more as blackpowder would. One of the more exaggerated effects (beyond gunky buildup) is ultimately smoke. The bombs listed above would create a lot of black smoke and likely provide smoke cover when detonated as well. This tactic could be expanded on by putting the saltpeter in a line across a field and lighting it. The result should be a relatively large smoke screen that could have relevance in army formations.
I believe it may be possible to create a bronze 'bombard' style weapon as well. A large bronze 'pot' could be filled with saltpeter and the opening to this pot hold a large iron ball. Igniting the pot (and running like ****) if lucky, would launch the iron ball a decent distance (little to no accuracy here) and make a feasible siege weapon. Unlucky sees this entire setup explode, sending shards of the pot everywhere. Probably a one shot weapon as the explosion would likely warp the bronze pot pretty heavily even in the event it worked.
This also would allow for a anti-wall technique that could make traditional siege unnecessary. Borrowing the scene from Lord of the Rings : Two towers, it would not be infeasible to mine under an enemy wall, load the entire tunnel full of this early explosives, and detonate the entire tunnel causing the walls to collapse as the earth underneath gives way.
If any of this is effective, it could heavily impact bronze age warfare. We are all decently familiar with the phalanx image (from 300 at very least) which sees a large number of men in a very tightly packed formation...this happens to be the best targets for gunpowder style bombs. Potential results:
1- Horses need training. Explosions are scary when you're not used to them...war horses would need exposure to these explosions long before battle.
2 - Move away from Phalanx and towards looser more mobile formations. Not good if an entire Phalanx unit can be dropped by one bomb.
3 - Move away from armour. Bronze age forces included exceedingly heavy Armour, almost to the point of silliness. Armour does not fare well vs gunpowder (piercing shards concentrate the impact on a small area, poking holes and allowing the shards to fly in). Edit as per comments - Bronze plate isn't the same as medieval plate...it's movement are restrictive and joints have gaps. Lower legs and arms were often unprotected (for movement reasons). A shrapnel weapon would play havok on anyone in one of these bronze suits.
I'm actually thinking, if the bombs were in any way effective, you'd see skirmish tactics over take the use of formation tactics
All speculation, but it could be a heavy heavy impact.