Bloodlines are traits passed down through your ancestors. They have different effects, including added regeneration, special bloodline jutsu, and various passive effects.
Depending on how powerful they are (how useful they are to strengthening your RPC), bloodlines are assigned a rank - the lowest being D, the highest being S. Some (always lower ranked) bloodlines may have only adverse effects while some (usually higher ranked) bloodlines may have only positive effects.
There are two ways to get a bloodline: rolling and buying.
Once you reach Genin rank you will have one chance to roll a bloodline for free at the Old Cabin. You may or may not get a bloodline from this roll, and if you do get a bloodline it will be random.
If you get a bloodline you don't want, you can remove it by buying an Etched Stone from the Black Market.
If you don't get a bloodline, or want a different bloodline, you can buy a bloodline-containing item from the Black Market with reputation points. Different items will give you different ranks (e.g. a Ritual Dagger will give you an A-rank bloodline), but the rank is the only thing you can choose. Even if you roll for an A-rank bloodline with a Ritual Dagger, for example, you will still get a random A-rank bloodline.
The only exceptions are S-rank bloodlines, which can only be gained by a lucky roll.