Suppose I wanted to send a fax at the lowest supported baud rate.
Well even if you have the modem part working a fax takes a lot of memory to hold the mage so that won't happen on an AT85.
I should also add that beyond the basic handshaking that's needed to sustain a connection, I don't really want to receive anything.
But that handshaking is receiving something so your system would have to have the receive capabilities.
sending a string of characters to a remote computer's terminal window
That is a lot simpler but you will need something at the computer end to act as a demodulator. Look at something like a radio ham's RTTY system, that is about as simple as it comes. RTTY from it:-
The transmitting part of the modem converts the digital signal transmitted by the teleprinter or tape reader to one or the other of a pair of audio frequency tones, traditionally 2295/2125 Hz (US) or 2125/1955 Hz (Europe). One of the tones corresponds to the mark condition and the other to the space condition. These audio tones, then, modulate an SSB transmitter to produce the final audio-frequency shift keying (AFSK) radio frequency signal.