We report new Northern Extended Millimeter Array (NOEMA) observations of the CII, NII and OI atomic fine structure lines and dust continuum emission of J1148+5251, a z=6.42 quasar, that probe the physical properties of its interstellar medium (ISM). The radially-averaged CII and dust continuum emission have similar extensions (up to θ=2.51+0.46−0.25 arcsec, corresponding to r=9.8+3.3−2.1 kpc accounting for beam-convolution), confirming that J1148+5251 is the quasar with the largest CII-emitting has reservoir known at these epochs.Moreover, if the CII emission is examined only along its NE-SW axis, a significant excess (>5.8σ) of CII emission (with respect to the dust) is detected. The new wide–bandwidth observations enable us to accurately constrain the continuum emission, and do not statistically require the presence of broad CII line wings that were reported in previous studies. We also report the first detection of the OI and (tentatively) NII emission lines in J1148+5251. Using Fine Structure Lines (FSL) ratios of the [CII], NII, OI and previously measured CI emission lines, we show that J1148+5251 has similar ISM conditions compared to lower–redshift (ultra)-luminous infrared galaxies. CLOUDY modelling of the FSL ratios exclude X–ray dominated regions (XDR) and favours photodissociation regions (PDR) as the origin of the FSL emission. We find that a high radiation field (103.5−4.5G0), high gas density (n≃103.5−4.5cm−3) and HI column density of 1023cm−2 reproduce the observed FSL ratios well.