The intensity and the linear scattering polarization profiles of the hydrogen Lyα line encode valuable information on the thermodynamic and magnetic structure of the upper layers of the solar chromosphere. The Chromospheric Lyman-Alpha Spectro-Polarimeter (CLASP) sounding rocket experiment provided unprecedented spectropolarimetric data of this line, as well as two-dimensional broadband images in intensity and linear polarization. We theoretically investigate the potential of the Lyα broadband polarimetric signals for probing the solar chromosphere and its magnetic fields. We analyze the synthetic Stokes profiles obtained from a series of radiative transfer (RT) calculations out of local thermodynamic equilibrium, considering semi-empirical one-dimensional models of the solar atmosphere. The wavelength-integrated linear polarization signal is found to be dominated by the contribution from the wings when considering a Gaussian weighting function with a FWHM that corresponds to the CLASP slit-jaw broadband filter. These broadband linear polarization signals are strongly sensitive to magnetic fields of strengths on the order of 50 G, via the action of magneto-optical (MO) effects, and are expected to encode information on the middle-upper chromosphere. The two-dimensional broadband intensity and linear polarization images observed by CLASP can be suitably mimicked using synthetic wavelength-integrated signals obtained considering atmospheric models and magnetic fields that are representative of solar regions with different levels of activity, provided that the impact of MO effects is taken into account. Despite the limitations of a one-dimensional RT modeling, this work illustrates the diagnostic potential of filter-polarimetric Lyα signals for probing the solar chromosphere and its magnetism.