  We present Keck/MOSFIRE $H$-band spectroscopy targeting C$\,$III]
$\lambda$1907,1909 in a $z=7.5056$ galaxy previously identified via Ly$\alpha$
emission. We detect strong line emission at $1.621\pm0.002\,\mu$m with a line
flux of ($2.63\pm0.52$)$\times10^{-18}$ erg s$^{-1}$ cm$^{-2}$. We tentatively
identify this line as [C$\,$III] $\lambda$1907, but we are unable to detect
C$\,$III] $\lambda$1909 owing to sky emission at the expected location. This
gives a galaxy systemic redshift, $z_{sys}=7.5032\pm0.0003$, with a velocity
offset to Ly$\alpha$ of $\Delta$v$_{Ly\alpha}$ = $88\pm27$ km s$^{-1}$. The
ratio of combined C$\,$III]/Ly$\alpha$ is 0.30-0.45, one of the highest values
measured for any $z>2$ galaxy. We do not detect Si$\,$III]
$\lambda\lambda$1883, 1892, and place an upper limit on Si$\,$III]/C$\,$III]
$<$ 0.35 ($2\sigma$). Comparing our results to photoionization models, the
C$\,$III] equivalent width (W$_{\text{CIII]}} = 16.23\pm2.32\,$\AA), low
Si$\,$III]/C$\,$III] ratio, and high implied [O$\,$III] equivalent width (from
the $Spitzer$/IRAC [3.6]$-$[4.5]$\simeq$0.8 mag color) require sub-Solar
metallicities ($Z\simeq0.1-0.2 Z_{\odot}$) and a high ionization parameter,
log$\,$U $\gtrsim -1.5$. These results favor models that produce higher
ionization, such as the BPASS models for the photospheres of high-mass stars,
and that include both binary stellar populations and/or an IMF that extends to
300 M$_{\odot}$. The combined C$\,$III] equivalent width and [3.6]$-$[4.5]
color are more consistent with ionization from young stars than AGN, however we
cannot rule out ionization from a combination of an AGN and young stars. We
make predictions for $James~Webb~Space~Telescope$ spectroscopy using these
different models, which will ultimately test the nature of the ionizing
radiation in this source.
