URI's are very common in kittens. And doxycycline is often very effective for treating them. Kittens with URI's often have very irritated nasal membranes. And yes, they can have a bloody nose simply due to the infection or the sneezing that goes along with the infection. However, kittens can have more serious problems causing bloody noses. Trauma is very common in kittens... they play rough with other pets or even by themselves. They can also get foreign material up their nose. Dust is very common and usually does not pose a long term problem. Other things, such as plant or grass-like material can lead to serious infections in the nasal passage. It is rare for a kitten to develop a polyp in the nose, but it could happen. If your kitten is rubbing at his nose or if you see swelling on one side or the other of the bridge of the nose or if the bleeding continues, this could indicate a more serious problem and should be checked by a veterinarian right away. If your kitten seems completely normal otherwise, and the bleeding has stopped, then you want to simply keep an eye him.
Some kitties with URI's will benefit from steam. Steam from an air humidifier or even from the bathroom shower can help moisturize irritated nasal passages and help open up clogged sinuses (both help the kitten breathe easier). Obviously, do not put the kitten in the shower, just let him run around the bathroom while the shower is on. Warm steam is the best.