Aims. We started a follow up investigation on the 'Deep X-ray Radio Blazar Survey' objects with declination > -10 deg for a better understanding of the blazar phenomenon. We undertook a survey with the European Very Long Baseline Interferometry Network at 5 GHz to make the first images of a complete sample of weak blazars, aiming at a follow-up comparison between high- and low-power samples of blazars. Methods. We observed 87 sources with the EVN at 5 GHz along the period October 2009-May 2013. The observations were correlated at the Max-Planck-Institut fuer Radioastronomie and at the Joint Institute for VLBI in Europe. The correlator output was analysed using both the AIPS and DIFMAP software packages. Results. All of the sources observed were detected. Point-like sources on milli-arcsecond scale are found in 39 cases, 45 show core-jet structure, and 3 are triples. The total ux density distribution at 5 GHz has a median value <S>=44+23-10 mJy. A total flux density ~150 mJy is observed in 68 out of 87 sources. Their brightness temperature Tb ranges between 10^7 K and 10^12 K. According to the spectral indices previously obtained with multi-frequency observations, 58 sources show a at spectral index, 29 sources show a steep spectrum or a spectrum peaking at a frequency around 1-2 GHz. 14 flat spectrum plus 1 steep spectrum sources are associated to gamma-ray emitters detected by the Fermi Gamma-ray Space Telescope. Conclusions. We performed the first milli-arcsecond resolution observations of a sample of suggested blazars. We found that among the 87 objects under investigation, 56 can be considered blazars. We also detected 2 Flat spectrum Narrow Line Radio Galaxies. 14 blazars plus a Steep Spectrum Radio Quasars, adding those observed with the EVN to those with ATCA oservations, are associated to gamma-ray objects. About 50% of them are BLLacs confirming they are more preferably detected among blazars gamma-emitters. We confirm the correlation found between the source core flux density and the gamma-ray photon fluxes down to fainter flux densities. We also found that weak blazars are also weaker gamma-ray emitters compared to bright blazars. 22 sources are SSRQs or Compact Steep-spectrum Sources and 7 are GigaHz Peaked Sources. The available X-ray ROSAT observations allowed us to suggest that CSS GPS quasars are not obscured by large column of cold gas surrounding the nuclei. We did not find any significant difference in X-ray luminosity between CSS and GPS quasars.