  We present SCExAO/CHARIS high-contrast imaging/$JHK$ integral field
spectroscopy of $\kappa$ And b, a directly-imaged low-mass companion orbiting a
nearby B9V star. We detect $\kappa$ And b at a high signal-to-noise and extract
high precision spectrophotometry using a new forward-modeling algorithm for
(A-)LOCI complementary to KLIP-FM developed by Pueyo (2016). $\kappa$ And b's
spectrum best resembles that of a low-gravity L0--L1 dwarf (L0--L1$\gamma$).
Its spectrum and luminosity are very well matched by 2MASSJ0141-4633 and
several other 12.5--15 $M_{\rm J}$ free floating members of the 40 $Myr$-old
Tuc-Hor Association, consistent with a system age derived from recent
interferometric results for the primary, a companion mass at/near the
deuterium-burning limit (13$^{+12}_{-2}$ M$_{\rm J}$), and a
companion-to-primary mass ratio characteristic of other directly-imaged planets
($q$ $\sim$ 0.005$^{+0.005}_{-0.001}$). We did not unambiguously identify
additional, more closely-orbiting companions brighter and more massive than
$\kappa$ And b down to $\rho$ $\sim$ 0.3" (15 au). SCExAO/CHARIS and
complementary Keck/NIRC2 astrometric points reveal clockwise orbital motion.
Modeling points towards a likely eccentric orbit: a subset of acceptable orbits
include those that are aligned with the star's rotation axis. However, $\kappa$
And b's semimajor axis is plausibly larger than 75 au and in a region where
disk instability could form massive companions. Deeper $\kappa$ And
high-contrast imaging and low-resolution spectroscopy from extreme AO systems
like SCExAO/CHARIS and higher resolution spectroscopy from Keck/OSIRIS or,
later, IRIS on the Thirty Meter Telescope could help clarify $\kappa$ And b's
chemistry and whether its spectrum provides an insight into its formation
environment.
