Does this mean I should strengthen the floor with plywood first?
Only if it needs it.
If in doubt, here's a system which I trust to beef up floors.
After removing top layers of underlayment and vinyl etc ... install 5/8" exterior grade plywood over existing subfloor and then install Ditra membrane over top the plywood, or you may use cement board. After that you're ready to tile.
An extra layer of plywood will remove the deflection (up/down movement) of the subfloor between the joists, and will help distribute the load over more joists but you may still need to add blocking between your joists depending on how bad your floor is.
