I have a Nikon D5000 with one Yongnuo RF-603II trigger. How can I use my Yongnuo YN-568EX flash as an external flash? What settings do I use?
If you want to use the RF-603II to trigger the flash as a transmitter, return/sell the YN-568EX flash, and get a YN-685 or YN-560III or IV, or if you want to use it as a receiver on the flash, buy a RF-603, RF-603II, RF-605, or YN-560-TX to act as the on-camera transmitter.
The YN-568EX has no built-in radio receiver. And to get a radio connection, you need both a transmitter and a receiver. The YN-568EX only has optical sensors in it to be used as a CLS slave in Nikon's proprietary wireless system (which requires a D7000 or above, or a commander flash unit on the hotshoe), or "dumb" optical triggers which can be set off by the D5000's pop-up flash (S1 = if the popup is in M mode; S2 = if the popup is in iTTL).
If you do get another RF-603II-compatible trigger to use as the on-camera transmitter, make sure that the YN-568EX is out of all the slave modes, and in M mode. The RF-603II is a manual-only trigger, so you can't use TTL/HSS, or remotely change the power on the flash from the camera. If you want these features, you'll have to consider the YN-622N triggers, and upgrading your camera body to one that can do HSS (FP flash).
You need a flash with a built-in radio receiver that understands the RF-603II. The two speedlight models I listed above both have this capability. Otherwise, you have to add a radio receiver unit that speaks the same protocol as the RF-603II onto the foot of the YN-568EX.