When I run my leg hurts in the back, a little under my calf. Is this my running shoes or something serious?

No way to tell. If the pain persists you should be seen by a doctor to determine the cause.
May have injury. Most, non traumatic, running injuries/discomfort are secondary a combination of: overuse, foot function/structure, shoes and training surfaces. See your doctor for an evaluation and discussion. I would recommend you stop for fitness activities, stay in good shoes and supports, antiinflammatories as tolerated until your doctor visit. Dr. L.
Could be serious. You should see your doctor. From the area described it could be tendinitis, strain of the lower gastroc/soleus, or even a stress fracture.