These photos show a rather extreme example of a dirty carrier which hasn’t been cleaned in about 5000 rounds.
For cleaning a carrier I like the Carbon Scraper Tool and Wipe Out Tactical Advantage Cleaner (see links below to purchase). Here I just simply applied the Tactical Advantage and used a toothbrush. You can also soak the carrier in this solvent to make carbon removal even easier as it won’t harm the chrome lining. Any places you can’t reach with the Carbon Scraper Tool, you can use a razor blade to carefully remove thick carbon deposits. But if you don’t go 1000 rounds or more without cleaning your carrier you won’t have this much carbon to scrape off. Also note that you can use the part of the scraper used for the bolt tail to clean carbon from the retaining pin area of the firing pin.