Privately or publicly. You have a choice of dealing with some or all of your questions as a group or on an individual basis. However, some questions might be fun for your players to collaborate on and discuss together. For example, perhaps the PCs are all family members, or the group's common origin might be decided if it wasn't established at the beginning of the campaign. Mandatory or optional.
Many players show up just to thrash the monsters and collect the gold.
A character questionnaire is the last thing they want to do. To avoid upsetting these players, feel free to make your surveys voluntary. If someone handed you one hundred questions to answer about a fictitious person, you might become a little intimidated. Ask just a few questions at a time, or one to three per session. Use some multiple choice and true or false questions.