  Monolayer PtSe$_2$ is a semiconducting transition metal dichalcogenide
characterized by an indirect band gap, space inversion symmetry, and high
carrier mobility. Strong intrinsic spin-orbit coupling and the possibility to
induce extrinsic spin-orbit fields by gating make PtSe$_2$ attractive for
fundamental spin transport studies as well as for potential spintronics
applications. We perform a systematic theoretical study of the spin-orbit
coupling and spin relaxation in this material. Specifically, we employ first
principles methods to obtain the basic orbital and spin-orbital properties of
PtSe$_2$, also in the presence of an external transverse electric field. We
calculate the spin mixing parameters $b^2$ and the spin-orbit fields $\Omega$
for the Bloch states of electrons and holes. This information allows us to
predict the spin lifetimes due to the Elliott-Yafet and D'yakonov-Perel
mechanisms. We find that $b^2$ is rather large, on the order of $10^{-2}$ and
$10^{-1}$, while $\Omega$ varies strongly with doping, being about $10^{3} -
10^{4}$\,ns$^{-1}$ for %typical Fermi levels in the interval $(10-100)$ meV,
carrier density in the interval $10^{13}-10^{14}$\,cm$^{-2}$ at the electric
field of 1 V/nm. We estimate the spin lifetimes to be on the picosecond level.
