It's pretty easy to knock up a satellite phone system and with any simple computer connected to it it can transmit your lat/long, plus any data which your avionics is capable of squirting out as RS232, in the form of an SMS message.
One gotcha is that satphone SMS is pretty unreliable. A lot of messages are simply never delivered. I have no idea why, or whether this is only on a specific satphone network.
Another one is that the avionics data is not generated in the first place. With modern avionics you could do a fair bit, especially if you have an
air data computer, which acts as a data concentrator, taking data from GPS, heading, airspeed, OAT, fuel flow, and emits a single data stream with all these parameters in it. If you can get an avionics shop to install a connector somewhere on which this data comes out, then the rest is not hard to do.
Other than SMS, you could do it using plain old dial-up internet access. But neither of these is going to be cheap on the phone billing - you need to bank on somewhere of the order of $1 per transmission.
I would wonder who is supposed to be receiving the data! It would be much simpler to just log it aboard the aircraft. Unless you plan to die enroute... but then you want to transmit pretty frequently
to somebody who is sure to be monitoring it.