hey
Theres quite a lot to talk about context wise! So for a detailed description check these out
They helped me out a bunch.
http://www.sparknotes.com/shakespeare/tempest/context.htmlhttps://www.cliffsnotes.com/literature/t/the-tempest/about-the-tempestA good one to discuss is colonialism, as it directly relates to Caliban's character. The Tempest, and Shakespeare in general, is rather progressive in it's critique of colonialism and imperialism which is interesting
In terms of how much you should talk about, don't go too crazy with it (which is exactly what I tend to do :/ haha), as you should be analysing the text in relation to discovery, not just providing a narrative of the time. However it's good to slip some stuff in every so often to demonstrate wider knowledge and understanding, especially if it helps to prove a point you are making.
I'm probably not the best person to ask in regards to integration, but just every so often (usually at the beginning of a paragraph/essay) provide 1-2 sentences of contextual detail? Theres not much else I can say in that regard, but I good way to practice is to write an essay the way you would normally, then go back and look over it, and see if there are any moments where a bit of context will help the sentence, and add it there!
Hope this helped! Good luck