After a rough rinse just under the tap - I use a doohickey I attach to the kitchen tap so I get a thin fast stream of water from the hot tap (via a short hose) - I use that to rinse off all the goop, the thin fast stream allows me to get into all the nooks and crannies of the filter. I keep going till I cant see any more "surface" soiling.
Then I soak overnight in pbw, rinse repeat, rinse really really well, dry completely and store.
I used to just soak in nappisan.. but found that over time my filter performance dropped off, taking longer and longer to filter a batch. I figured it was because either pores were getting permanently blocked or that I was getting biofilm build-up and it was cutting down the effective size of the pores. Either way I figured PBW would help to fix it, so an overnight soak in hot pbw happened. Filter suddenly back up to full speed!! So now I use pbw. It sounds like a rampant product endorsement I know, but I think pbw is close to magic in how well it work on almost everything I have tried it on... wouldn't be without it in my brewery.
Before next use I run sanitiser from keg to keg through the filter.
Keg A full of sanitiser
Connect Keg A to filter, connect filter to Keg B
Connect C02 to Keg A
Push the sanitiser through the filter from Keg A to Keg B
When all the sanitiser is through, make sure you allow enough pure gas to run through to push all the sanitiser out of the filter too.
Keg B becomes Keg A for my next filter run.
Then you've sanitised your keg, completely removed all the air and oxygen, sanitised your filter, flushed it with C02 as well and you probably only use 100ml of sanitiser that you don't recover for next time.
I am a fair bit more fussy about it than many other people... but I think that you need to be with filtered beer if you want to get good shelf life. Consider me the anally retentive end of the scale when deciding what you want to do
TB