We report the experimental results of simultaneous measurements on the electron and X-ray spectra from near-critical-density (NCD) double-layer targets irradiated by relativistic femtosecond pulses at the intensity of 5E19 W/cm^2. The dependence of the electron and X-ray spectra on the density and thickness of the NCD layer was studied. For the optimal targets, electrons with temperature of 5.5 MeV and X-rays with critical energy of 5 keV were obtained. 2D particle-in-cell simulations based on the experimental parameters confirm the electrons are accelerated in the plasma channel through direct laser acceleration, resulting in temperature significantly higher than the pondermotive temperature. Bright X-rays are generated from betatron emission and Thomson backscattering before the electrons leave the double-layer targets.