  We study the formation of primordial black holes (PBHs) from density
fluctuations due to supercooled phase transitions (PTs) triggered in an
axion-like particle (ALP) model. We find that the mass of the PBHs is inversely
correlated with the ALP decay constant $f_a$. For instance, for $f_a$ varying
from ${\cal O}$(100 MeV) to ${\cal O}$($10^{12}$ GeV), the PBH mass varies
between $(10^{3} - 10^{-24}) M_{\odot}$. We then identify the ALP parameter
space where the PBH can account for the entire (or partial) dark matter
fraction of the Universe, in a single (multi-component) dark matter scenario,
with the ALP being the other dark matter candidate. The PBH parameter space
ruled out by current cosmological and microlensing observations can thus be
directly mapped onto the ALP parameter space, thus providing new bounds on
ALPs, complementary to the laboratory and astrophysical ALP constraints.
Similarly, depending on the ALP couplings to other Standard Model particles,
the ALP constraints on $f_a$ can be translated into a lower bound on the PBH
mass scale. Moreover, the supercooled PT leads to a potentially observable
stochastic gravitational wave (GW) signal at future GW observatories, such as
aLIGO, LISA and ET, that acts as another complementary probe of the ALPs, as
well as of the PBH dark matter. Finally, we show that the recent NANOGrav
signal of stochastic GW in the nHz frequency range can be explained in our
model with $f_a\simeq (10~{\rm GeV}-1~{\rm TeV})$.
