  We characterize the performance of a gravimeter and a gravity gradiometer
based on the $^{1}$S$_{0}$-$^3$P$_0$ clock transition of strontium atoms. We
use this new quantum sensor to measure the gravitational acceleration with a
relative sensitivity of $1.7\times10^{-5}$, representing the first realisation
of an atomic interferometry gravimeter based on a single-photon transition.
Various noise contributions to the gravimeter are measured and characterized,
with the current primary limitation to sensitivity seen to be the intrinsic
noise of the interferometry laser itself. In a gravity gradiometer
configuration, a differential phase sensitivity of 1.53~rad/$\sqrt{Hz}$ was
achieved at an artificially introduced differential phase of $\pi/2$~rad. We
experimentally investigated the effects of the contrast and visibility based on
various parameters and achieve a total interferometry time of 30~ms, which is
longer than previously reported for such interferometers. The characterization
and determined limitations of the present apparatus employing $^{88}$Sr atoms
provides a guidance for the future development of large-scale clock-transition
gravimeters and gravity gradiometers with alkali-earth and alkali-earth-like
atoms (e.g., $^{87}$Sr, Ca, Yb).
