Getting 5V Output on FT245RL
Contrary to what you wrote, SJ1 is not between VCCIO and VCC. It selects whether the FT245 chip gets IO-side driver power from the built in 3.3V regulator (default) or from the target device via JP3 pin VCCIO. 

So you must either leave VCCIO connected to the 3.3V regulator via jumper SJ1 (which results in max 3.3V output swing), or cut SJ1 and connect VCCIO to some other supply, perhaps from the external target system, or from USB. You have done the latter, so that's promising.

So two further points:
1. Check that 5V is actually getting to the VCCIO pin using a voltmeter. If it is, then...


Note that the output pins have only limited ability to deliver current, and the more current needed by the external load, the lower will be their max voltage in the high state. Spec sheet says Voh typical 4.1V and min 3.2V with a load of only 2mA. 


Only if the outputs are unloaded will you see close to 5V. That's fine if your external system has high-impedance inputs, as, for example, Arduino inputs would be. But if you are testing with an external system connected but powered down, then those external connections will present a significant load due to their input protection diodes.  

Likewise if you are using these outputs to drive LEDs or even transistors (with base resistors) you will see lower output voltages.

So... test with external system disconnected.

Schematic: https://www.sparkfun.com/datasheets/BreakoutBoards/FT245RL-Breakout-Schematic-v2.pdf
Data sheet: http://www.ftdichip.com/Support/Documents/DataSheets/ICs/DS_FT245R.pdf