Which chapters does Tonari no Kaibutsu-kun anime cover? Which chapters does the anime cover? Where should I start reading in order to continue from the anime?
The anime covered about the first 4 volumes of the manga. 

If you want to pick up the manga then starting at about chapter 16 would be a good choice. 

If I remember correctly there was a little left out in the final episode of the series, but nothing critical.