  We present a detailed study of the electronic and spin-orbit properties of
single and bilayer graphene in proximity to the topological insulator
Bi$_2$Se$_3$. Our approach is based on first-principles calculations, combined
with symmetry derived model Hamiltonians that capture the low-energy band
properties. We consider single and bilayer graphene on 1--3 quintuple layers of
Bi$_2$Se$_3$ and extract orbital and proximity induced spin-orbit coupling
(SOC) parameters. We find that graphene gets significantly hole doped (350
meV), but the linear dispersion is preserved. The proximity induced SOC
parameters are about 1 meV in magnitude, and are of valley-Zeeman type. The
induced SOC depends weakly on the number of quintuple layers of Bi$_2$Se$_3$.
We also study the effect of a transverse electric field, that is applied across
heterostructures of single and bilayer graphene above 1 quintuple layer of
Bi$_2$Se$_3$. Our results show that band offsets, as well as proximity induced
SOC parameters can be tuned by the field. Most interesting is the case of
bilayer graphene, in which the band gap, originating from the intrinsic dipole
of the heterostructure, can be closed and reopened again, with inverted band
character. The switching of the strong proximity SOC from the conduction to the
valence band realizes a spin-orbit valve. Additonally, we find a giant increase
of the proximity induced SOC of about 200%, when we decrease the interlayer
distance between graphene and Bi$_2$Se$_3$ by only 10%. Finally, for a
different substrate material Bi$_2$Te$_2$Se, band offsets are significantly
different, with the graphene Dirac point located at the Fermi level, while the
induced SOC strength stays similar in magnitude compared to the Bi$_2$Se$_3$
substrate.
