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Those things all depend on the personality of your character and the setting, I would think. For example, if your character comes from the middle ages where disease and famine are very common and many laws are punishable by death, they might not be taken off guard as much if they lose a parent. They will surely still be upset, but in those days death was all around and common. If you think about current times, because of medicine and advancements and such things, deaths are usually only by accident or incurable diseases. Because of this they aren't expected nearly as much, and the fact that you don't expect somebody to die can make it much harder to deal with when they do.

As for personality, think of how your character reacts to other things. (This [link] would probably help you a lot in figuring out how your character reacts to things).

For parental/family death, a character's reaction would probably be affected by things like how close they were to the parent, how strongly it impacts their life aside from the actual death, and who else the have in their lives. If they live with their mother and their father dies, they might be upset by it but it would probably be a bit easier to get over because of the lack of connection before the death. However, if they lived with both their parents and were very close to them, if both parents were to die they would have to deal with not only the fact that their parents are gone, but the fact that their entire life would change. They would probably have to leave their home to live with another relative, perhaps even going as far as another state or country. In that case you have to consider that they might be forced to leave pets, friends, or significant others behind. Dealing with the death of a family member would be a lot harder if you didn't have any friends or people you were close to to help you through it.

As for overreacting, just try to fit it in to the character's reaction. If they have a bad day, do they kick a wall and cuss a lot? Or do they just bottle it up inside and answer "fine" when someone asks how their day was? The first would probably blow up and possibly do something drastic like run away or break something if they found out they were adopted. The second might run out of the house, but come back a few hours later and slip into their room to sulk in silence. Now, if your character really loves his adopted parents, he will surely be quicker to forgive them or accept it than if he had problems with his parents. If he really didn't like his parents, he might try to seek out his real ones to find a better life. Also, not all people would be devastated by such a realization. Perhaps they always expected it, or maybe they love their adoptive parents so much that they consider them more real than any other parents could ever be. Additionally, consider the age at which the character finds out. If they find out when they're very young, it will be easier to accept than if they find out when they're into their teens or early twenties. Having your parents keep such a big secret for that long creates a huge sense of betrayal and mistrust that could make the reaction a lot more severe.

To combine the two, if a character were to find out their adopted and try to seek out their real parents, then find that their real parents were dead, their reaction could range greatly. If they had been trying to hunt down their real parents for years and years, meeting such an abrupt end to their search could upset them greatly. If they had only just begun the search, it might be easier to get over. It would also change depending on how they feel toward their adopted parents.

Sorry that was super duper long and I hope it helped some!