One of the operational costs for a satellite in orbit is maintaining a receiver ground station. Would it be feasible to send an Iridium transceiver along with my satellite, and then operate it from my personal computer via the Internet?
Yes, if your satellite will orbit in Low Earth Orbit (LEO) below Iridium Constellation's orbital altitude (about 780 km), then this could work as has been experimentally demonstrated by the “Bell” PhoneSat 1.0 launched in April, 2013. But you better complete all the paperwork before the launch not to share the same fate as the TechEdSat that was supposed to also use Iridium frequencies but its operators had to disable that module and only keep the Orbcomm SatPhone module because they didn't get both its frequency ranges approved by FCC in time. From the latter linked to source:
“We were forced to disable the Iridium modem as our FCC license did not come in time. As usual, building the satellite is the easy part.”