Harry, Ron and Hermione, while hiding, try to decipher the clues left to them by Dumbledore and to destroy Voldemort's Horcruxes after he had took over the Ministry of Magic. I am quite doubtful about the quality of the question. But I am answering the question best with my knowledge and understanding
A)
Harry, Ron and Hermione, while hiding, try to decipher the clues left by Dumbledore to them and to destroy Voldemort's Horcruxes after he had took over the Ministry of Magic. - "he had took" has two problems - 1. "had" does not agree with "took" 2. "had" is needed for the earlier of the two events in the past
B)
Harry, Ron and Hermione, in hiding, try to decipher the clues left to them by Dumbledore and to destroy Voldemort's Horcruxes after he takes over the Ministry of Magic.[/color] - This is the "best" of the options available, but this sentence still is not correct in all ways. "He" should have ideally referred back to "Voldermort", but instead it refers back to "Voldermort's Horcruxes" since "Voldermort's" is possessive. For a possessive noun, only a possessive pronoun is used. The other antecedent for "he" is "Dumbledore", which is logically incorrect (PS - Not a big Harry Potter fan!)
C)
Harry, Ron and Hermione, while in hiding, tries to decipher the clues left to them by Dumbledore and destroy Voldemort's Horcruxes after he took over the Ministry of Magic. - "Tries" is a singular verb and refers back to "Harry, Ron and Hermoine", which is plural
D)
Harry, Ron and Hermione, while in hiding, try to decipher the clues left by Dumbledore to them and destroy Voldemort's Horcruxes after he takes over the Ministry of Magic. - Other than the issue mentioned in B, D also says "destroy" instead of "to destroy"
E)
Harry, Ron and Hermione, while in hiding, tried to decipher the clues left to them by Dumbledore and to destroy Voldemort's Horcruxes after which he took over the Ministry of Magic. - "which" incorrectly modifies "destroy Voldermort's Horcruxes"
Since this is not an official question and we do not know the source of the question, I'd not bother about this sentence a lot!
GMATNinja - Could you please provide your comments as well. Thanks!