Batteries made for use in RC racing cars are designed to deliver high current with low voltage drop. To do this they have relatively thick plates. High capacity batteries use thinner plates to leave more room for active material, which increases their resistance and reduces the maximum discharge current. Depending on how much current the motor in your car draws you may get away with using high capacity NiMH cells, but you probably won't get the run time you would expect based on their stated capacity.
Cheap high capacity consumer cells often have a low maximum discharge current and are not tolerant of abuse. Many of them do not achieve their stated capacity at any current because they are 'optimistically' rated. However they may still be better than the original cells in your car (which wouldn't be using AA cells if it was designed for high performance!).
Before deciding which batteries to buy you should examine independent test data to determine whether they will be suitable. Here is a website which has test data for several popular AA rechargeable cells, whose performance can be compared at various discharge currents. Choose cells which hold their voltage up best at high current (5-10A) rather than selecting for highest capacity.
Building your own pack is possible, but unless you have access to a spot welder you will have to solder the cells together. This must be done quickly to avoid cooking the cells (particularly the positive end which has a plastic separator that must not be allowed to melt). First clean the battery contacts with fine sandpaper, then immediately 'tin' them using a high wattage iron and rosin-cored solder (should take no longer than 1 second). For the connections between cells I use thin desoldering braid saturated with solder.
Your existing charger will probably work fine, but will take 2-3 times longer to charge the higher capacity battery. Chances are that it is a simple 'trickle' charger that just delivers a fixed current with no shutoff, which is safe provided the expected charge time is 10 hours or more. If it is a more sophisticated 'peak detect' charger (designed to charge the original battery in about 1 hour) then it may not be suitable because NiMH batteries have a different charging profile.