the book John suggested is great, I recommend to look through it once at least.
For my experience, Michael-Type Addtion click chemistry is rather simple to handle (we use it frequently to synthesize e.g. PEG-Peptide conjugates). You need a free cystein SH-Group at the peptide linker and a dye with a maleimide group. Sigma Aldrich offers a ton of maleimide dyes (e.g. Atto dyes). The cystein due to its acidic character will start to deprotonate at pH above 4.5/5. If you use e.g. PBS at pH 7.4, the two components (-S(-) of Cys and Maleimide) will react for sure - the beauty behind that is that you do not need any fancy solvents, (toxic) activation reagents like EDC/NHS or HOBt/TBTU, no extra crosslinker and there are no side products (thats one of the reasons why it is called Click chemistry).
I attached a link where you find extra information about the chemistry I described. They specify the reaction conditions, purification of the compounds, possible side reactions with impurities etc.