Problem with making an Arduino-like board
First, clean the board thoroughly with an appropriate solvent such as 99% isopropanol or a petroleum based solvent, removing all flux and look at it under a microscope or magnifier. Fix anything that looks dodgy. Use at least 5x magnification, enough that you can see the solder balls in the paste if they are not melted. Tilt the board to and fro to catch anything not visible from directly above.
Since you've already powered it up, try that again and check that power supply voltages exist everywhere they should. I suggest using the USB power source. Be extremely careful not to short the probe when checking lest you add to the woes.
At this point your FT232 should show up as a COM port on a PC if you have the correct driver installed. Get that working first, only a few parts are associated with the USB-to-serial converter. If there is a hardware problem it's likely to be associated with the power or the two USB signal wires.