  The inherent asymmetry of the electric transport in graphene is attributed to
Klein tunneling across barriers defined by $\textit{pn}$-interfaces between
positively and negatively charged regions. By combining conductance and shot
noise experiments we determine the main characteristics of the tunneling
barrier (height and slope) in a high-quality suspended sample with Au/Cr/Au
contacts. We observe an asymmetric resistance $R_{\textrm{odd}}=100-70$
$\Omega$ across the Dirac point of the suspended graphene at carrier density
$|n_{\rm G}|=0.3-4 \cdot 10^{11}$ cm$^{-2}$, while the Fano factor displays a
non-monotonic asymmetry in the range $F_{\textrm{odd}} \sim 0.03 - 0.1$. Our
findings agree with analytical calculations based on the Dirac equation with a
trapezoidal barrier. Comparison between the model and the data yields the
barrier height for tunneling, an estimate of the thickness of the
$\textit{pn}$-interface $d < 20$ nm, and the contact region doping
corresponding to a Fermi level offset of $\sim - 18$ meV. The strength of
pinning of the Fermi level under the metallic contact is characterized in terms
of the contact capacitance $C_c=19 \times 10^{-6}$ F/cm$^2$. Additionally, we
show that the gate voltage corresponding to the Dirac point is given by the
work function difference between the backgate material and graphene.
