What happens in and around cells during inflammation, on a biochemical level?

Inflammation response on the individual cell level does not make a lot of sense. It is a tissue-level phenomenon. On the individual level some cells get apoptotic, other cell types get recruited to the inflamed area, some have cytoskeletal re-arrangements to allow that to happen, etc. It all depends on the affected tissue, the respective cell type and so on.

In the roughest form you can see it as a response to tissue injury in which a lot of unspecific responses are activated to eventually restore tissue integrity.