As far as I understand, to gain merits through combat for any power, you go to any opposing faction and kill certain ships in the system. Which npcs do I kill and where do I find them? Through interdiction or nav beacon or conflict zone?
Rating 3 requires 750 merits. Do I need this amount of merits or is there competition to reach that amount?
Also, do I need to wait 1 power cycle to raise rating?
the first thing you should do is to double check that its really just ships you need to kill to undermine, as this can vary a little depending on the power you are attacking.
to do that select the power you want to attack in the power rating screen and go to their control tab, there you can read what you need to do.
once you have verified what it is you need to do you should go to one of their control system, either select one from that tab or go to map and select the power you wish to attack and click the "control" check box.
if you also want your attack to have any impact on PP you should pick a control system that is not 100% fortified.
once there you find the ships in super cruise, usually spawning near the star and traveling towards the stations. so set yourself up somewhere along that path to interdict them.
you get merits for all ships that are flagged as "power" ships, even power ships that does not belong to the faction that owns the control system.
Rating 3 requires 750 merits. Do I need this amount of merits or is there competition to reach that amount?
no competition, all you need is your 750 merits.
Also, do I need to wait 1 power cycle to raise rating?
yes.