You'll likely want to start off with a survey so you can get to know the historiography and general outline of the British Empire. The preponderant historian at the moment on British imperialism is probably John Darwin, so anything by him (Unfinished Empire, The Empire Project) will set you in good stead. Failing that Bernard Porter is a safe bet.

Cain & Hopkins' British imperialism, 1688-2000 is notable for introducing the idea of 'Gentlemanly Capitalism', which while isn't fully accepted is an important concept to familiarise yourself with. They also wrote an article on the subject, so the book can be forgone if you so desire (though it's still a worthwhile read)

Probably the most important thing you can read on British Imperialism is the work of Robinson & Gallagher (good British Empire historians come in twos, evidently). They outlined their theory of 'Informal Empire' in Africa and the Victorians: the official mind of imperialism, and followed this up with an academic article on the subject that is probably easier to grasp ('The Imperialism of Free Trade'). Most books will likely go over the historiographical debate on informal empire but it's probably worth your time to read that original debate as it's incredibly important to fully understand the concept of informal empire, even if you ultimately disagree with it (note that the debate goes back further than R&G to Fay and Hancock, but that should be explained in any survey).