First, you require a dirty gun. You should not drop one in the garden or the vacuum cleaner bag; you need one dirty from shooting; not only is this the proper way to get your gun dirty, it's a lot more fun. And noisy.
Now you need cleaning supplies. Oil and/or bore cleaner, cleaning patches, a cleaning rod, the jag to hold your patches, and a large flat space to lay things out on. This surface should not be one that is easily stained or damaged, or should be one you can hide. If you're married, you already know why(if you're not, you'll find out one day)
First, make sure the firearm to be cleaned is unloaded, with no cartridge left in the chamber. Not doing this could be noisy, and lead to more damage-cover problems.
Second, take it apart enough to clean it. There are two levels of this:
Field-stripping means taking it apart just enough to clean things off.
Detail-stripping means taking the whole damn thing apart; barrel, screws, stock, springs and all. This method is much more fun, allowing you to clean everything and giving much more opportunity to lose things and practice your language skills. Especially if you've lost the directions to put it back together.
For most needs, field-strip the thing and lay the parts out in order.
Remove the cat from the table.
Wet a patch with your chosen cleaning agent and wipe things off, setting them down in order. Put a patch on your cleaning rod, wet it down, and run it through the bore. Repeat, and set aside.
About this time you'll find that either a: the cat has returned to the table, or b: you're out of patches. Take a moment to either throw the beast off or dig out some more patches. You don't have more? Find that old t-shirt and start cutting. By the time you finish this, you're thirsty, so you have to wash your hands and get a drink. Do wash first; gun oil/cleaner does not help the taste of any drink known to man. And put the bottle down, you can have that after you're done.
Return. If you detail-stripped the piece, you'll likely find that either the cat came back to visit and is batting things around, or you bumped the table and things rolled around. This is your first opportunity to practice your language skills. Which, if you have kids, may also get you a lecture you really don't need right now.
Take a rag and wipe all the parts off, hopefully taking off all the fouling. Then wipe them down again with a clean patch & oil. Push a couple of clean patches through the bore, hopefully wiping all the crud out of the bore. If not, run a couple more wet patches through and wipe out again.
You should also have a couple of spare rags handy, or a roll of paper towels. This is to wipe up the spill when you knock the bottle over. You will do this either when a: you reach for it, or b: when the !*(@)#*&^^@ cat jumps back up on the table. If yours is a really fiendish beast, it WILL jump up where reaching for it will cause you to knock the damn bottle over.
Wipe up spill, and mutter language lesson, then run clean patches through bore. Unless you're shooting really crappy ammo or have let it get really nasty, this should do it.
Wipe all excess oil off all parts, and reassemble. And you're done!
Unless, of course, you detail-stripped it. Then you get to look for all the parts that have rolled/fallen somewhere, and try to find your directions(which you then get to try to open up without getting oily fingerprints all over it, causing further display of verbal skill). After finding all necessary parts/materials, then you get to put it all back together.
And now, flush with success, you get to put the firearm away, head back to the sink to wash your hands off- followed by washing out the sink so you don't receive a language lesson.
There are variations on this basic routine. Usually involving parts not being found, falling over the cat/dog, and sometimes ending with screamed imprecations at the pet, the parts involved, and possibly the ancestry of whoever designed the damn thing in the first place.
Now is the time to retire to a chair with a bottle of suitable drink.
Wasn't that easy?