Can I enter the Church privately to avoid upsetting my family?
I am convinced in my heart that I must become a Catholic, but my family is troubled, and my becoming Catholic could lead to divorce. I wish the priest at the local church could let me quietly enter the Catholic faith so that this could be avoided.
If necessary, your local priest can receive you into the Church privately. That is often done for those who are in situations where a public reception would pose a grave familial or financial problem for the person. However, he would want to be assured that the stakes really are high and that waiting a reasonable length of time would not change them.
If waiting a reasonable length of time to be received could prevent grave sins from being committed by others, then waiting could be permitted. However, it could not alter the basic decision to convert, since we are bound by divine law on that point, and it would be a grave sin for a person who has recognized the truth of the Catholic faith not to join the Church.
The fact that you have recognized the truth of the faith thus settles the question of whether to convert, but the circumstances of one's life help settle when and how the reception into the Church is to be done (immediately or after a period, at Easter or at some other time, publicly or privately, etc.).
What counts as a reasonable length of time also varies according to a person's situation. What is a reasonable length of time for someone in a normal situation is not the same as a reasonable length of time for one on his deathbed.
However, a "reasonable length of time" is not an indefinite period that can stretch on for years with no end in sight. Once a reasonable period has elapsed, one is bound to go ahead and enter the Church, whether this is done publicly or privately.