The paper reports the results of two-dimensional particle-in-cell simulations of proton beam acceleration at the interactions of a 130-fs laser pulse of intensity from the range of 10<sup>21</sup> – 10<sup>23</sup> W/cm<sup>2</sup>, predicted for the Extreme Light Infrastructure (ELI) lasers currently built in Europe, with a thin hydrocarbon (CH) target. A special attention is paid to the effect of the laser pulse intensity and polarization (linear - LP, circular - CP) as well as the target thickness on the proton energy spectrum, the proton beam spatial distribution and the proton pulse shape and intensity. It is shown that for the highest, ultra-relativistic intensities (~ 10<sup>23</sup> W/cm<sup>2</sup>) the effect of laser polarization on the proton beam parameters is relatively weak and for both polarizations quasi-monoenergetic proton beams of the mean proton energy ~ 2 GeV and δE/E ≈ 0.3 for LP and δE/E ≈ 0.2 for CP are generated from the 0.1-μm CH target. At short distances from the irradiated target (< 50 um), the proton pulse is very short (< 20 fs), and the proton beam intensities reach extremely high values > 10<sup>21</sup> W/cm<sup>2</sup>, which are much higher than those attainable in conventional accelerators. Such proton beams can open the door for new areas of research in high energy-density physics and nuclear physics as well as can also prove useful for applications in materials research e.g. as a tool for high-resolution proton radiography.