Below are the links to two scientific papers that address different ways to modify aspects of lysine’s chemical structure. Because lysine has a primary amine group, it can undergo a lot of possible reactions (mostly with electrophilic compounds, as primary amines are strong nucleophiles). The paper by Spicer and Davis goes into detail on how this is possible—it also discusses the modification of cysteine, in case you’re looking for something to compare lysine modification to. The other paper describes the modification of lysine residue with citraconic anhydride in detail in the experimental section.
These methods could show the reactivity of lysine with different compounds. More information can be obtained from the resulting product depending on the kind of characterization tests performed (NMR and FTIR analysis are both great ways to identify the various functional groups that are present in a molecule). Hopefully this gives you some ideas!