Displaying ASCII codes to an 7x5 LED Matrix
There is no best way without specifying constraints.  
Assuming you are only making 1 unit, the BOM cost is likely second
after feature flexibility.
Embedded processor provide flexibility far greater than combinational logic.


Normally, LED matrix displays such as these are not all tied to 1 common cathode.  Rather, they are arranged in a matrix:



Most embedded processors have enough programmable GPIO pins to drive such a display.  Here is a project in which an Atmel embedded processor (Arduino) is used to drive an 8 x 8 LED matrix.

Once built, you can program the embedded processor to display your ASCII data in any number of ways.  As 5 binary 7 bit numbers in each of the 5 vertical columns.  As a graphical character as defined by this table:



Since an embedded processor is being used, you imagination is free to create any pattern to represent your data.