@strangebedfellows said:

Can any one enlighten me please ? How did they know the bones belonged to the original Red?

They have the DNA of the original Reddington on file. Harold Cooper tested the blood from the shirt held in evidence to determine that Liz is Red's daughter. That shirt was taken from Red when he was a Naval officer and was tortured by the Russians, and rescued by a SEAL team. Somehow they never tested Spader's DNA it seems. But when they tested the bones and got DNA which matched the DNA on file with the Navy they knew the bones belonged to Red Reddington. So they then realized that Spader was not the real Reddington. But apparently Lizzie has not yet told Cooper about the DNA results of the bones yet.

I wonder if Spader might be Lizzie's uncle on her mother's side. One time he told Brian Dennehy something to the effect "I know you're disappointed in the choices I made, that I didn't turn out the way you wanted". That sounds like something a son might say to a father. And Dennehy is Katarina's father. So that would make Spader her brother, and Lizzie's uncle.