I did Chekov for my exam last year. Loved his work. Even though i didn't do Russian Revolution or have a Russian mum, i spent hours researching his background, contemporary Russia, reading documents/reports/essays on his style of writing compared to other styles, etc. So this research put me in a pretty good perspective about what his stories may have been about. Of course, his writing is subjective and concise, so you can't help but wonder if there's always more to what he's writing on the surface.
Anyway, it's good to include a little bit of the history of Russia; but be aware that Chekov wrote in a time just before the Revolution, so focus much more on the understated social tensions between the common people, especially. Make sure that in your inclusion of the Russia's history that you don't recount what happened historically - instead, put it in perspective of his stories and how this 'calm before the storm' affected his characters. In particular, i noticed that many of his characters didn't seem too have much energy or enthusiasm for life: dull, resigned, ineffectual. But here's the question that you have to ask: why did Chekov write about these boring people and their boring lives?
Also, attached is just some notes that i gathered from the interwebs after countless hours of research. Enjoy!