Usually the bore of a shotgun doesn't get that dirty. Most of the crap simply wipes out with a patch or a "mop". Most shot loads are contained in plastic "wads" which keep the shot from touching the barrel metal. Just go buy a generic $10-20 shotgun cleaning kit (Hoppe's is a good brand) with a sectional cleaning rod, bronze bristle brush, a jag with which to run large cotton patches through the bore, and a "mop" to finish up with.
Believe it or not, I use motor oil. Mobil1 0W-30 to be specific. And I use cheap tooth brushes and Q-tips to wipe out the crud from the small places. Then I use clean Q-tips to apply the oil to just about every interior surface where moving parts rub together. Just a thin coat of oil mind you, not globs or enough that it starts to run and drip. Just quickly dip the Q-tip into the oil and pull it back out. Let the oil completely soak into the Q-tip and then proceed to re-oil the guns surfaces. Don't forget to occasionally wipe the outside of the gun with an oily cloth too in order to prevent rust.
Why motor oil? Simple. If it works in a car engine with tolerances that are FAR smaller than in any gun, and at the speeds and number of cycles that an engine produces, then it'll work just fine in a gun that is looser and whose parts move much slower. Motor oil also contains detergents which are great at dissolving and removing crud and powder residue.
Being a synthetic oil, it also stays fluid at really freaking cold temperatures and yet still "sticks" to the parts at really hot temps.
And the best part is that a $5 quart of Mobil1 will last you many, many years.
Eric