How does Hamlet's upbringing affect his initial response to the task of killing Claudius?
In Shakespeare's day, Revenge Tragedies were incredibly popular plays that followed a pretty formulaic structure -- Someone has been killed villainously and his death must be revenged for the world of the play to be set right again. The Elizabethans loved Revenge Tragedies and flocked to see them performed the way many people today flock to Action or Horror movies.
Hamlet is a take-off on a typical Revenge Tragedy. I say "take-off" because Hamlet does not behave in the conventional manner of a hero in a Revenge Tragedy. He questions the task itself. This would have come as quite a surprise move to his very savvy audience. Hamlet's pausing to question the veracity of the Ghost was a unique twist to the classic Revenge plot, added by Shakespeare.
And so we come to your question. If, by "upbringing," you mean his religious upbringing, then his upbringing plays a huge role in his questioning of the Ghost. As a Catholic, Hamlet would have been taught that there are good spirits (sent on mission of a divine nature) and bad spirits (sent from the Devil to do humans harm), and it becomes Hamlet's job to decide which of these the Ghost is. He says in Act I, scene iv, upon seeing the Ghost for the first time:
Angels and ministers of grace defend us!
Be thou a spirit of health or a goblin damn'd,
Bring with thee airs from heaven or blasts from hell,
Be thy intents wicked or charitable. . .
. . .I will speak to Thee.
So, Hamlet is aware of the possibility, from the first moment of seeing the Ghost,that it might not be his "father" come back from the dead at all, but some work of the Devil.
The question of the good or evil of the Ghost's intention, however, does not seem to affect Hamlet's initial response to the task. In Act I, scene v, he tells the Ghost, "Rest, rest, perturbed spirit," and finishes the act with a line that seems to indicate that he has taken up the task of revenging his father's murder. He says:
The time is out of joint. O cursed spite,
That ever I was born to set it right.
It is not until his "Now I am alone" soliloquy at the end of Act II, scene ii that he lets the audience know that he needs more proof in order to go forward with the revenge. He says,
. . .The spirit that I have seen
May be a devil. . .I'll have grounds
More relative than this. The play's the thing
Wherein I'll catch the conscience of the King.
For more on Revenge Tragedies and Hamlet's dilemma concerning the good or evil intent of the Ghost, please follow the links below.