"Bloodline" is an idiomatic way to refer to someone's family history.
In this case, "blood" is used to refer indirectly to genetic heritage (although the term predates the discovery of genetics). We can talk about being "related by blood" to someone as compared with "related by marriage". We might say some person's talent is "in his blood" to mean that others in his family have similar talents.
You child can be referred to as "your own flesh and blood" (although this can also apply to other relatives).
Another expression is "blood is thicker than water", meaning to have stronger obligations to family than to friends or strangers.
If someone has an older relative who has some character trait (usually a bad one), and people suspect they have it as well, they might say, "blood will tell" (or "blood will out"). This implies that the trait "runs in the family" and will eventually express itself.
As in your example, "bloodline" can also refer to animals, particularly those with a "pedigree" who may have been bred for generations to reinforce certain traits.