A .35 Rem probably does not have severe carbon fouling like a fast varmit rifle could. But - there could be a fair amount of copper down in the grooves, mixed with other odds and ends.
So, I recommend the following:
1. Scrub with a NEW brush, wet with a typical solvent (say Hoppe's), NOT copper solvent. Give it 10 or 20 strokes.
2. Couple of dry patches through the barrel. They will be filty, but some of this is the brush.
3. Couple of wet patches - but not solvent. Rubbing alcohol or something that will degrease the barrel. If you don't like the ideal of rubbing alcohol, then take it outside, and squirt some brake cleaner down the bore and let it run out and evaporate. I use denatured (99%) alcohol.
4. Now, wet a patch with some Sweet's or Barnes CR10. Push patch through barrel. When you get to the muzzle, add a little more solvent with a dropper or the like. I hold the gun upright, then put a little puddle of solvent in the end of the barrel with the patch about a quarter inch back from the muzzle. Pull patch out of barrel while keeping it vertical, then stand it muzzle-down on a rag or towel. The idea is to make sure the bore has no grease or oil, then completely coat it with copper solvent.
5. Let soak per the instructions on the bottle, then run a dry patch through. If it is green/blue, repeat process! If not, you do not have copper fouling.
Alternating the brush and copper solvent will help get fouling out that is in layers, that is, copper mixed with powder fouling and lead.
Note that by using a brush, you WILL always get some black gunk. The brush does wear quite a bit. Easy enough to prove, measure the diameter of the brush with a caliper before you start and after you are done. By the time you are done it will probaby be groove diameter, if not less.
Rinse brush off when done with HOT water. If overly worn, 'recycle' to the next smaller bore size (.35 to .338, .338 to .30, etc.). Or use to clean case necks.
If you are just going through endless cycles of the above, a patch wrapped around an udersized brush and coated with JB, or the Rem bore cleaner, will help speed the process.
Recognize that the copper solvents are typically water-based. Naturally, you will need to put some oil in the bore after using them.
Once you get the gun completely - and I mean completely - clean, you won't have to go through this circus each time. But it can really try your patience the first time you clean one.
Also, if this stretches over several days, leave Hoppe's in the barrel overnight, as it will prevent rust (says so right on the bottle).
If the gun continues to foul, then lap it, don't put up with the aggravation. The .35 Rem is easy, use .38/.357 wadcutters, and 3 grains of Bullseye. Marshall has lapping compound. 20 rounds of lapping bullets will do a rifle bore wonderrs.