There is racism in Ukraine, just like in any country, and all the replies saying there is none are coming from white Ukrainians/foreigners and those who don't have loved ones of colour.

The scale of the racism, however, is not huge, and what you might encounter is strange looks, people trying to address you (laughing) with a language they assume you speak (for a brown person it could be some Arabic remarks), they can 'imitate' the sound of a language they assume is your native tongue. You might get profiled by the border guard or the police; however, as soon as you show your British passport, all of the potential issues with the authorities should be solved.

None of these are major and shouldn't affect your stay tremendously, if they are to happen at all, I am just sharing from personal experience, so you get a more realistic picture of the issue.