Nightwing and Batman visit Joker in Arkham to see whether he was related to the Red Hood in any way, which he is not. However, his actions caused Jason Todd's death and subsequent transformation into Red Hood after his resurrection, so in a way the Joker was linked to Red Hood in more ways than one.
Red Hood seems to use an awful lot of explosives, to the point of it becoming signature. Joker killed him with an explosion, and he probably got so traumatized by it, that it became his signature.
Fridge Tearjerker: Batman's no-kill rule to villains, as seen in this feature, is not by any means a sign of altruism or value of life, but it's the only thing keeping him from becoming as bad as the villains he fights. And sadly, Red Hood crossed that line with breaking that rule, no matter how desperate or successful his goals were.
After Joker finishes beating Todd to within an inch of his life, he turns back as he's leaving and tells him, "Please tell the big man I said... hello." At first, it seems he's referring to Batman. But after it's revealed that he's rigged the place to explode long before Batman can rescue him, it takes on a new meaning altogether. He's not referring to Batman; he's referring to God.