Two cabins with connecting doors one with twin beds with pullmans to sleep 3 and the other for Mum and Dad. This gives you two showers/toilets. Choose an inside cabin or outside on lower decks to save costs.
Whomever you travel with I can't see you getting a room/suite to accommodate five so you will have to book two cabins. As for Royal Caribbean they have family suites which will take four in twins and bunks. They are expensive and won't allow for five. Their normal cabins theoretically accommodate up to four guests but, as there are only two beds I don't know how.Also, unless a cabin is designed for a larger number of people, like the suites you would be tripping over each other.
Other companies, such as P&O,and I speak from experience here have some three berth cabins which would solve your problem. You could book two cabins next to each other; (interlocking ones are available if you book early enough) one two berth and one three berth or even two three berths if they are available (these always go fast) and they would not charge you extra for two people in a three berth. The only thing you need to be aware of is that it is a condition of booking that there will be an adult in each cabin. Of course if you are happy to leave the kids all sleep together then no-one is going to be any the wiser but,of course their safety and security in those circumstances would be entirely at your own risk. It just depends how old your kids are and whether or not you can rely on them to behave.
Whatever you book, have a great time. Cruising is by far the best way to holiday.