The design and synthesis of new biocompatible and biodegradable copolymers for biomedical applications is one of the current challenges. This study presents the synthesis of graft copolymers consisting of oppositely charged polysaccharide and polypeptide, namely, chitosan or heparin as the main chain and poly(L-glutamic acid) or poly-L-lysine as the side chains. A metal-free click chemistry based on the azide-alkyne 1,3-dipolar cycloaddition reaction has been used to synthesize the graft copolymers. The copolymers are characterized by 1H NMR and FTIR spectroscopy, size-exclusion chromatography, and quantitative high-performance liquid chromatography. The obtained graft copolymers are able to form nanoparticles in aqueous media. Depending on the composition of the graft copolymer and conditions, the polymers form spherical nanoparticles of 200–420 nm with a positive charge of 33–54 mV. In contrast to the initial polycation, the graft copolymers are found to be less toxic to human embryonic kidney cells (HEK 293T). The obtained systems are able to efficiently bind nucleic acids at an optimal copolymer/pDNA ratio of 50–100 while maintaining a positive charge. The developed copolymers demonstrated the ability for intracellular delivery of GFP pDNA into HEK 293T cells with an efficiency of up to 71% depending on the copolymer and conditions.