Building a .service file turned out to be overkill. After briefly growing to a huge mess, my final Dockerfile looks like this:
FROM vmware/photon
MAINTAINER sam.denton@emc.com
ADD http://influxdb.s3.amazonaws.com/influxdb-0.9.3-1.x86_64.rpm my.rpm
RUN rpm --install my.rpm --noscripts
RUN ln -s /opt/influxdb/versions/0.9.3/influx /usr/local/bin/.
EXPOSE 8086
CMD ["/opt/influxdb/versions/0.9.3/influxd"]
The "ln" seemed easier than adding the installation directory to my PATH, and lets me run the administrative CLI via docker exec. There's still a couple of tweaks to make, like setting up a VOLUME to hold the database, but this seems to be working well for me.