This paper discusses the use of a novel amphiphilic graft polymer poly{(lactic acid)-$co$-[(glycolic acid)-$alt$-($L$-glutamic acid)]}-$g$-monomethyl poly(ethylene glycol) (PLGG-$g$-mPEG) as a drug carrier. PLGG was synthesized through the ring-opening polymerization of $L$-lactide (LLA) and (3s)-benzoxylcarbonylethyl-morpholine-2,5-dione (BEMD) using $Sn(Oct)_2$ as a catalyst and it was subsequently deprotected $via$ hydrogenolysis in the presence of Pd/C. A series of monomethyl poly(ethylene glycol) (PEG) with the molecular weights of 2,000, 1,100, and 500 were immobilized on the carboxyl groups of PLGG. These PEGylated graft derivatives were characterized using proton nuclear magnetic resonance spectra ($^1H$ NMR), Fourier transform infrared spectroscopy (FTIR), and gel permeation chromatography (GPC). The critical micelle concentrations (CMCs) of the amphiphilic copolymers were tested by the fluorescence probe technique and the CMCs were 2.3, 1.0, and $0.32{mu}g/mL$, respectively. Transmission electronic microscopy (TEM) and dynamic light scattering (DLS) images revealed that the micelles were homogeneous spherical nanoparticles and the sizes of the micelles were distributed across a range of 80 to 22 nm. Anticancer drug doxorubicin (DOX) was loaded into the micelles. The $in$ $vitro$ release profiles showed that the sustaining release of the drugloaded micelles could last over 7 days. The $in$ $vitro$ cytotoxicity assay of the DOX-loaded micelles against HepG2 cells was assessed by methyl thiazolyl tetrazolium (MTT) assays. The results demonstrated that the drug-loaded micelles exhibited a high level of inhibition activity on cancer cells. The confocal microscopy images of HepG2 cells showed that DOX released from the micelles could be delivered into cell nuclei. PLGG-$g$-mPEG micelles are promising potential carriers for delivering anticancer drugs.