Check the power play menu in the Nav Panel. If you are pledged to Sirius Corp, and go to a foreign powers screen, you will see a tab with a list of their control systems. If you select a control system, it will tell you how to undermine that system in the middle of that screen (but only after you have pledged to a competing power). Usually, you undermine by killing ships that are hauling power play cargo for that power. When you travel to that system, the targets will be labeled as either private system security or delivery agents of that specific power. When you kill either target type you will now accrue 30 merits which you can verify under your transactions tab on your Nav Panel. In order to cash in those merits however, you need to travel back to your own control system and go to your power contact, and verify the kills via the missions screen. This is like turning in a normal mission at a station, but instead of doing it through the bulletin board, you do it through the contacts menu.
You do not spend any money to do this, except in repairs and ammunition. The only way to spend money to gain ranks in PP is by Fortification. This is basically just a mail delivery simulation. You get packages to deliver to various stations, which are limited to a certain quantity per hour based on your rank. You can pay to speed up the process by skipping the timer for each package at a cost of 10,000 CR per package. This adds up to 100M CR to gain rank 5. And then a maintenance amount based on how quickly rank degrades. They modified the decay formula recently though, so I couldn't tell you the exact amount you need to grind each week, but at most it's 50M CR to rebuy rank 5 every week.
ninja'd
Many thanks to Ziljan and others for the info. I've spent some time (while nursemaiding my Asp back towards the bubble) reading about PP and I'm a lot more informed now.
I reckon I'm carrying about Cr70M (based on my tally of explored stuff) of exploration data, and that will take me to Elite, so that is a bit more precious than the extra credits from Sirius Corp that, as others have correctly stated, I can make somewhat easier in other ways.
I think I'll just cash in at the first opportunity and then decide whether to dabble in this PP thing...
Cheers all!