Activated charcoal in toothpaste may help remove surface stains on your teeth. Charcoal is mildly abrasive and is also able to absorb surface stains to some degree. There is no evidence, though, that it has any effect on stains below a tooth's enamel, or that it has a natural whitening effect.
For deeper stains the underlying colour of your teeth will not be improved by charcoal toothpaste, a major lightening of tooth colour can only come from professional whitening treatments.
But it’s very important to use an approved activated charcoal toothpaste. The best charcoal toothpaste products should be free of things like sodium lauryl sulfate erythritol but should contain Fluoride.
Don’t be too aggressive when using charcoal toothpaste - gently brush in a circular motion and use infrequently. Charcoal can be quite abrasive, and it’s very important to protect the tooth enamel to maintain a healthy smile.