wow. this is a really general question with a lot of assumptions for us to make - including your knowledge about electronics and stuff.
first of all: yes. it's doable. in principal you can exchange every chemistry against any other. BUT you have know what you do.
lead acid is very user friendly, but has very little energy density. very heavy - low energy.
li-co/mn/... is very high power density, and lighter.
both have different charge/discharge properties. and cell voltage is different as well. a 6 cell lead is around 12v, a 6 cell lithium is around 20v.
your lead battery is 5x 12v battery, so 60v in theory. fully charged 14v * 5 = 70V. it's quite likely that your controller can take a little more than that.
for that you would need the correct charger. lithium should be "balance" charged. that means that during charge every single cell is monitored so it won't charge above (4.2-4.3V). if it did the cell could go mad and bad things can happen.
so 3 ways to do it (2 recommended):
.) monitor cell voltages during charge - no recommended as not practical and time consuming
.) use a balance charger - uncomfortable for anything above 12 cells
.) use a bulk charger and use a bms to monitor the battery cells.
so i'd recommend, depending on your knowledge:
a) get some 18s li-mn 18650 pack as big as you can with a proper bms and a bulk charger. imho by far the most convenient and safest setup.
b) get some 6s (maybe 3 in series) rc-lipo batteries and a rc-charger or bms. this is the most powerful, and cheapest setup, but bears the highest risk, especially if you are not used to work with this kind of battery.
question that remains: you didn't define power. do you want to acchieve more range, or more horsepower? for both you need to swap the battery first, so this was the first part of my answer.
for more power: you may want to post pics of the scooter and it's insides. controller, motor stuff etc ....