No; Death Jr. (or DJ, as he's known for short) was not created in a grotesque ritual where Death Sr. sliced open his own flesh and bled himself onto a fire. See, Death's actually a pretty nice guy, and is married to a nice (and normal, for all intents and purposes) woman. He has every intention of living a nice conventional life in suburbia (if Death can be said to be alive) with his wife and son, along with his continued career of collecting the souls of the departed.
A Gift and A Curse
Unfortunately, it's not actually that easy, especially for DJ. Born the eventual heir to the mantle of Grim Reaper, he had a hard time growing up because he couldn't control his powers to cause those around him to die. This had a tragic effect on his attempts to keep pets, and perhaps isolated him from other children initially, as well.
Even as his powers came more under control, DJ's skeletal appearance is probably a little disturbing to other children--as the Seep put it, "He looks like a walking pirate flag!" Fortunately, there are rather a lot of other weird-looking kids in his town, and hence they're not all that entitled to cast the first stones.
Worse, though, is the fact that his world is occupied by a great deal of beings just as powerful as his father, and a lot meaner--and DJ can rarely seem to stay out of trouble with them. Nonetheless, although he's still young and naive, Death Jr. is a chip off the old bones, and will genuinely try to do the right thing every time he figures out there's something very wrong with things.
Powers & Abilities
Death Jr. has the power to kill mortals by touching them, and in some cases, even affect the probability of events causing those near him to die. Earlier in his adventures, he had very little control of these powers and accidentally killed Pandora in the first story arc, although her ghost stuck around until her life was restored. By now, it appears DJ has mastered his more menacing abilities and his friends are quite safe around him.
Though he's still in training to be the Reaper, Death Jr. has already adopted the scythe as his primary weapon in combat, and wields it quite well. He also routinely carries twin pistols for long-distance combat, and a legion of hamsters trained as suicide bombers. He has also wielded multiple other guns. Normal children probably wouldn't be allowed so much heavy firepower, but DJ's parents obviously don't seem to worry that he'll accidentally take an eye out, perhaps because DJ seems not to have any eyeballs.