Issues after capacitor replaced on AC unit
If the lights are dimming several times while the unit is running, the only thing you can check is listen to the outdoor unit to determine if the compressor is unexpectedly starting and stopping when it should be running continuously.  Set the thermostat to a low setting and listen outside to see if you can hear the compressor starting and stopping at about the same interval as you notice the lights dim.

If abnormal starts and stops are occurring, a competent technician is needed to evaluate and repair the problem.  Whether or not the compressor is cycling unexpectedly, installation of an appropriately sized 3 wire hard start kit can sometimes alleviate lights dimming on startup due to the electric service being marginally small for the unit.

You may want to try a better technician than the one who advised an entirely new system and then replaced a leaking capacitor.  Any technician worth his salt would use a capacitance meter and not an amp meter to legitimately investigate this problem.  Also, unless your 6 year old system was installed absolutely horridly, it should not need replacement so soon.