Does a laptop computer charge faster when off?

No one answer fits all, it really depends on the laptop!

Some have built in features (vendor specific) that speed up the battery charging, others (including most of mine) charge a lot slower when in use.

Again, no one answer fits all.

It depends on the power supply. We have a few computers which can be run off a lower-wattage power supply while running - it will not charge the battery at all. If you attach a larger power supply, then it can charge and run.

Check with your laptop manufacturer. It seems somewhat common to see power supplies with different wattage being offered. The power supply with a higher wattage may be what you need to charge quickly and power the system while on. Check to see if a larger power supply is available if you need to charge quickly while using the system.