We report a $^{205}$Tl NMR study of vortex state for an aligned polycrystalline sample of an overdoped high-$T_c$ superconductor Tl$_2$Ba$_2$CuO$_{6+\delta}$ ($T_{c}\sim$85 K) with magnetic field 2 T along the c axis. We observed an imperfect vortex lattice, so-called Bragg glass at $T$=5 K, coexistence of vortex solid with liquid between 10 and 60 K, and vortex melting between 65 and 85 K. No evidence for local antiferromagnetic ordering at vortex cores was found for our sample.