I strained my back this weekend and have blood in my stool currently. Is this a common cause & effect?

No. Sprained back and blood in stool are not likely to be related, but they can certainly coexist. Some common causes of blood in stool would be hemorrhoids or anal fissure. But it can be due to other more serious causes. So it wound be prudent to get checked by a physician.
? NSAIDs. I would also be concerned about blood in the stool if you have been using large amounts of nsaids (ibuprofen, alleve, etc..) as that may put you at risk for a GI bleed. Usually it will be a proximal source such as an ulcer, in which case the stool is more commonly black. Either way you need to see a physician especially if it continues.