  (Abridged) In this paper, we present analyses of images taken with the
Herschel ESA satellite from 70mu to 500mu. We first constructed column density
and dust temperature maps. Next, we identified compact cores in the maps, and
characterize the cores using modified blackbody fits to their SEDs: we
identified 684 starless cores, of which 199 are bound and potential prestellar
cores, and 132 protostars. We also matched the Herschel-identified young stars
with GAIA sources to model distance variations across the Perseus cloud. We
measure a linear gradient function with right ascension and declination for the
entire cloud. From the SED fits, mass and temperature of cores were derived.
The core mass function can be modelled with a log-normal distribution that
peaks at 0.82~$M_\sun$ suggesting a star formation efficiency of 0.30. The
high-mass tail can be modelled with a power law of slope $\sim-2.32$, close to
the Salpeter's value. We also identify the filamentary structure of Perseus,
confirming that stars form preferentially in filaments. We find that the
majority of filaments where star formation is ongoing are transcritical against
their own internal gravity because their linear masses are below the critical
limit of 16~$M_\sun$pc$^{-1}$ above which we expect filaments to collapse. We
find a possible explanation for this result, showing that a filament with a
linear mass as low as 8~$M_\sun$pc$^{-1}$ can be already unstable. We confirm a
linear relation between star formation efficiency and slope of dust probability
density function and a similar relation is also seen with the core formation
efficiency. We derive a lifetime for the prestellar core phase of
$1.69\pm0.52$~Myr for Perseus but different regions have a wide range in
prestellar core fractions, hint that star-formation has started only recently
in some clumps. We also derive a free-fall time for prestellar cores of
0.16~Myr.
