A new water-soluble thermosensitive star-like copolymer, dextran-graft-poly-N-iso-propilacrylamide (D-g-PNIPAM), was created and characterized by various techniques (size-exclusion chromatography, differential scanning calorimetry, Fourier-transform infrared (FTIR) spectroscopy, and dynamic light scattering (DLS) spectroscopy). The viability of cancer cell lines (human transformed cervix epithelial cells, HeLa) as a model for cancer cells was studied using MTT and Live/Dead assays after incubation with a D-g-PNIPAM copolymer as a carrier for the drug doxorubicin (Dox) as well as a D-g-PNIPAM + Dox mixture as a function of the concentration. FTIR spectroscopy clearly indicated the complex formation of Dox with the D-g-PNIPAM copolymer. The size distribution of particles in Hank’s solution was determined by the DLS technique at different temperatures. The in vitro uptake of the studied D-g-PNIPAM + Dox nanoparticles into cancer cells was demonstrated by confocal laser scanning microscopy. It was found that D-g-PNIPAM + Dox nanoparticles in contrast to Dox alone showed higher toxicity toward cancer cells. All of the aforementioned facts indicate a possibility of further preclinical studies of the water-soluble D-g-PNIPAM particles’ behavior in animal tumor models in vivo as promising carriers of anticancer agents.