Although checking the pulse as a measure of health goes back at least as far as ancient Greece, the credit for figuring out how to measure blood pressure goes to Stephen Hales of England, in the 1700's. Hales actually began his studies by measuring the fluid pressure inside the vascular tissues of trees, which he published a paper on in 1727. In 1733 he did experiments in which he measured the blood pressure of a horse by inserting a thin brass tube into an artery and measuring how far up the tube the blood was pushed by the horse's heart.
It wasn't until 1855 that a way to measure blood pressure without opening a blood vessel was developed by Vierordt. You can browse thruogh pictures of the different devices at this link.
The understanding that blood is a "life fluid" in the body was something known to ancients. They understood in their limited way that there was an organ (the heart) that pumped blood through the body. The blood was pressurized by the heart to cause it to circulate, and this fact was known to people who killed and cut up animals for food. Given these facts, there is no one person who "discovered blood pressure" as was asked. As investigators probed the anatomy of animals and also that of the human body, our understanding of the circulatory system grew. And it continued to advance to become the part of modern medicine that it is today.