I have an enlarged right testicle with a heavy chord that extends upward from the top of the testicle. Could this be a hernia?

Maybe. If it is a recent occurance, it may be a hernia. It may also be a hydrocele, varicocele, infection, etc. It should be evaluated by yout doctor.
Hydrocele. Sounds like a hydrocele, possibly communicating with the abdominal cavity. An inguinal hernia could also present in this manner. A varicocele on the right is extremely unusual and requires immediate evaluation. Because of the anatomy, varicoceles are almost always located on the left. A right varicocele in a young otherwise healthy male is abnormal and requires immediate attention.
Varicocele. Likely the process is a varicocele but all other options should be ruled out. Urgent evaluation by your md or a urologist is necessary.