Hey GM - it's a regime but it's something I actually enjoy! I concur with the above post and clean 'properly' every 4-500 rounds however if I've been to the range I've invariably shot that so it's every time!
Like you said, after 200 odd rounds or so, the thing starts getting dirty whatever rounds you push through it. Bore snake is fine on its own if you've been out hunting and maybe only fired a dozen rounds or so - when I've been to the range, the following is my particular routine that works for me.....
Let the gun cool for at least 15 minutes before you start anything on the barrel - I strip the upper from the lower and give the lower a good blowout with compressed air (I use the kennair canisters you can get from a camera shop) this works well in blowing out all the loose stuff first - I also give the upper a blowout too. I take out the bolt and charging handing and then give the bolt a good spray with Browning Legia and put it to one side to 'soak'.
Then, on the upper I go in with cotton wool buds and an old toothbrush (I'll spray a little browning Legia on these to get things moving). Learnt the cotton wool bud thing off TacDaddy on YouTube (really good video on how to clean the 15-22 - highly recommend watch).
I keep going with these until they start coming out clean (can take a while as the receiver gets very gunked-up.
Once the upper (barrel) is back to ambient temp, I go in with the bore snake, 2 to 3 times being particularly careful not to snag it on the extractor

I then get a rod and put a lambs wool mop on it - I then put drops of Wipe Out bore treatment and Wipe Out accelerator on said mop and and push it through once. I'll then leave the barrel for 10 minutes or so for this to work.
Whilst that's happening I strip the bolt, guides, spring etc and give the bolt face etc a good going over with the tooth brush and cotton wool buds (cue tips for our brother over the pond). Make sure this is all clean and then blow out again with compressed air. Reassemble, spray a little more Legia and wipe it off, then this bit's done.
Back to the barrel, I pop a nylon brush on and push through half a dozen times - then on with the jag and push through maybe 8-10 patches until I'm happy with it being clean enough!
Upper and lower internals get a quick spray of Legia and wiped out - bolt assembly in and upper and lower put back together. Barrel and flash eliminator + thread gets some Clenzoil as my barrel suffers from what I can only describe as about 3 inches worth of 'parkerisation' (barrel goes grey basically) and the Clenzoil brings it back a treat......
And that's it, done, ready for the next time - it's a habit I've gotten in to with my other rifles and so the Smith gets the same love.....! Some may say it's overkill but I a. enjoy it and b. believe the gun will last longer for it....