Lysines are relatively easy to label at their epsilon-NH2, providing that you work at pH higher than 8 (ideally between 8.5-9.5). Coupling via EDC/NHS with any carboxy-dye (fluorescein, rhodamine...)  is pretty convenient and should grant you a good yield.This of course providing that you are labelling a long stretch of lysines. 

You may also want to think about a NHS/EDC chemistry on the N-terminal lysine or arginine.

If you can, however, order some pre-labelled peptides. Much cheaper than making the labelling yourself and much easier to purify.

Hi Natali, in addition to Marco's suggestions, you could do COOH coupling using a 1000-10.000 fold molar excess of a fluorophor that has a free amine function.

That way you are likely to get only one fluor on the -COOH terminus of the polymers and therefore directly compare poly-Arg and poly-Lys.

Just choose a bright fluor to label the -COOH terminus. Maybe a quantum dot.    

Lysine is most easily labeled in the unprotonated state.  From the perspective of neutral pH, the closer you are to its typical pK of 9.5 the more reactive it will be.  The N-terminal amine of a polypeptide has a pK of about 7.5.  Typically, people purchase NHS esters of the desired fluorophore (no EDC needed) or NHS activate a carboxyl derivative of the dye.  Typically we have used a 10- to 20-fold excess of dye and a pH closer to 7.5 to limit reactivity.  

NIHS esters will not modify Arginines.  Thus the N-terminal amine is a better target.  

If you are modifying polylysine and don't want every amine modified use pH close to neutral with limiting amounts of dye.

 Mr. Steingrimur, Please tell the proper method or link for using the COOH terminal of lysine to couple the Fluorescent dye having NH2 group free, if possible.

You can simply use commercial available NHS-Rhodamine in a buffer with 8.5 pH for labeling:

https://www.thermofisher.com/order/catalog/product/46406