What is the value of a 1963 five dollar US note?

In circulated condition, it's worth about $8

A nice crisp uncirculated one is worth about $15

A dealer will probably only pay face value to $6.

Serial Numbers

Note that a bill's serial number is just a counter so including it in a question generally doesn't help to identify it or affect its value. There are collectors who will pay a premium for bills with low or patterned serial numbers; e.g.12345678 or 00000005.