How do I display a certain character on a 16-segment display?
The chip you have shown is the right one. The important thing is to connect right pin to right segment -- this will obviously depend on the display yu have. 

For example, the top horizontal segment is labeled "a" in Figure 2. You will want to find your LED datasheet and find out which LED pin corresponds to "a" segment -- lets say it is pin 10 for example. Then you look at table 1, and connect pin 10 "a" of digit 0 (first digit) to pin 9 "O2" of MAX chip; then pin 10 "a" of digit 2 (third digit) goes to pin 7 "O0" of MAX chip, etc..  You want to get a pen and paper and very carefully plot the connection. There is an example layout in the evaluation kit: https://datasheets.maximintegrated.com/en/ds/MAX6955EVKIT.pdf

That said, it may be easier to skip using a custom chip and just go to general purpose microcontroller. This will require a bit more coding, but a much easier hardware design -- for example, you may be able to just wire all "a" pins together, "b" pins together, etc...