Yes,you can breed characteristics,If you give the Parent Pokemon with the characteristic you want a Power Item for the stat you want,the baby Pokemon has a chance of having that characteristic,Note that characteristics don't give the exact amount of IVs,they give a variety of numbers,e.g. the characteristic "Alert to Sounds" can tell you that your Pokemon's speed IVs may be on of these numbers 1, 6, 11, 16, 21, 26, 31. If you want a way to get good characteristics here's a good way,first you look for a Pokemon with the characteristic you want,Make sure that the IVs are 31(You can do an IV battle to get it's IVs)it doesn't have to be the pokemon you want to breed and train,just compatible (If not try chain breeding it or finding one that is compatible). then give that Pokemon a Power item that matches it's respective Characteristic you want to breed,then breed that Characteristic/31 IV to the Pokemon you want to train.
Here's a list of Characteristics and their IVs