Statistical and block copolymers based on poly(2-hydroxyethyl methacrylate) (PHEMA) and poly[oligo(ethylene glycol) methylether methacrylate] (POEGMEMA) were modified with 4-pentenoic anhydride or 4-oxo-4-(prop-2-ynyloxy)butanoic anhydride to generate polymers with pendant vinyl or acetylene, respectively. Subsequent thiol–ene or thiol–yne reaction with thioglycolic acid or 2-mercaptosuccinic acid leads to polymers with carboxylate functionalities, which were conjugated with cisplatin (cis-diamminedichloroplatinum(II) (CDDP)) to generate a drug carrier for Pt-drugs. Only the polymers modified with 2-mercaptosuccinic acid resulted in the formation of soluble well-defined polymers with gel formation being prevented. Due to the hydrophobicity of the drug, the block copolymers took on amphiphilic character leading to micelle formation. The micelles were in addition crosslinked to further stabilize their structure. Pt-containing statistical copolymer, micelles, and crosslinked micelles were then tested regarding their cellular uptake by the A549 lung cancer cell line to show a superior uptake of crosslinked micelles. However, due to the better Pt release of the statistical copolymer, the highest cytotoxicity was observed with this type of polymer architecture.