Once the bullet leaves the barrel of your gun, it has to sweep out a cylinder of air between the gun and the target. Air's about a thousand times less dense than lead, but by the time a 1cm long bullet has swept out a 10m cylinder it will have had to pass through approximately its own mass in air. Initially it will, in fact, collide with air particles, because it has a big fat cloud of electrons that will readily interact with air molecules. These will get blown off pretty quickly, hopefully leaving behind the nuclei which will be much more compact and will be able to travel much further before colliding with something
Each collision has so much energy that even nuclear binding forces are kinda negligible so you can easily end up with all sorts of interesting effects like fusion and fission as well as plain old deflection. The projectile has a lot of momentum, but those air atoms are still going to impart a small direction change to the bullet atoms. The bullet will be ablated away from the tip backwards, and the ablation will produce an expanding cone-shaped region of hot, bright plasma. It'll be filled with high-energy electrons, so it'll be a good source of beta radiation and x-rays, neither of which are good for the shooter or bystanders.
The projectile will sweep all the air away from in front of it, but will be travelling so fast that there's not really much time for the air to refill its wake before the bullet hits its target (or is ablated away). It might look a bit like a straight lightning bolt, and be pretty loud into the bargain when than channel collapses. Incidentally, a lightning bolt only carries the energy of a 5-50 microgram projectile travelling at .9c. Pretty weedy, eh?
By the time it reaches its target the projectile will be, at best, a dense blob of plasma, with some proportion of the nucleons it started out with. These particles don't "hit" the target in the sense of two solid objects hitting each other, but instead eventually get around into interacting with the target's atoms. The path lengths of heavy ions at .9c is a complex matter, but it will almost inevitably be longer than the target is thick. Overpenetration is therefore practically guaranteed. If your projectile is so small and light as to (hopefully) reduce the chance of this overpenetration, it will likely have insufficient mass and momentum for enough of it to make it to the target.
As the heavy ions decelerate at the target they'll release their kinetic energy by ionising and displacing atoms of the target, and by bremmstrahlung radiation. If the target is behind, or in front of, or carrying (or contains) dense metal components, this can result in some high-energy xrays being emitted, bad for bystanders and nearby scenery.
It isn't clear quite how much of the energy of the projectile will be left in the target as heat, but even 1-10% of a microgram-scale projectile will be enough to make them go bang with considerable force. The kinetic energy of a mere 20 micrograms at .9c is equivalent to the yield of a tonne of TNT. Your projectile would have to be nanogram scale in order not to produce problematic overkill.
Reducing the size of the projectile to limit overkill and radiation effects means your gun more and more closely approximates a GeV particle beam. Such weapons are a bit rubbish in air as they're too easily attenuated which limits their range and still releases hazardous radiation along their beam track. That's why the SDI program only considered them for use in a vacuum. Just use a laser, railgun or coilgun. You know it makes sense.
So, to recap:
It must not significantly risk the life of the shooter or innocent bystanders anywhere within 3 ft of the projectile's path or impact point.
It must not significantly risk incapacitating or injuring the shooter or innocent bystanders anywhere within 10 ft of the projectile's path or impact point.
The shooter can wear suitable protective gear to protect them from flash burns, x-rays and beta radiation. Bystanders cannot, but I suppose there's a reasonable chance that they'll survive and might not even suffer acute radiation sickness.
Anyone near the "impact point" as much as there can be such a thing, is probably hosed though. They'll end up in a (relatively diffuse, admittedly) plasma fireball and then the target will go bang with probably the force of a hand grenade, if not much more. They have a good chance of receiving a serious dose of x-rays, too. So this probably can't be achieved.
It must do maximum damage to the target following the above specifications.
Looks like damage is easy enough, but a gun which avoids the problems above probably can't actually shoot a target at anything other than point-blank range.