You can use also a cheap logic analizer, but you must know the voltage of the signals. Probably 3.3V, but what you measure with the dmm is the mean value. Normally the uart has 4 pins, gnd, vcc, tx and rx, perhaps the round contacts, You only need the gnd, that you can take of many places, of the ground plane, and the tx.
The logic analizer software can decode the messages if they are shorts, and discover the baudrate, then a uart to usb and a terminal program is better.