If the manual provided with Ukelele is not sufficient, you can ask for help with specific problems at the users group
https://groups.google.com/forum/?fromgroups#!forum/ukelele-users
It might be easier to just use the US Extended layout. Shortcuts for Turkish special characters are
option c, then c ç
option b, then g ğ
option w, then i ı
option u, then o ö
option c, then s ş
option u, then u ü
PS Have you tried the Turkish Qwerty layout (instead of Turkish Qwerty PC) in system prefs/language & text/input sources? It may also do what you need via the option keys while keeping the usual other shortcuts.
I had this exact same issue, and user Tom Gewecke's advice seems to have fixed it for me. I simply removed two plist files in ~/Library/Preferences, and then rebooted. The files were com.apple.CharacterPicker.plist and com.apple.CharacterPaletteIM.plist. I don't know which of those files had the issue, but after removing them and rebooting they've regenerated, and everything's good again.
If you're comfortable with Terminal
You probably know how to move files around, just move these two somewhere else, or rename them:
~/Library/Preferences/com.apple.CharacterPaletteIM.plist~/Library/Preferences/com.apple.CharacterPicker.plist
If you just want something you can copy and paste, here you go:
mv ~/Library/Preferences/com.apple.CharacterPaletteIM.plist{,-backup}
mv ~/Library/Preferences/com.apple.CharacterPicker.plist{,-backup}
Reboot after this, and fingers crossed, it will work for you again.
If you're not so comfortable with Terminal
Open up a Finder window and press Shift ⌘ G (Shift Command G), then type ~/Library/Preferences/, and press Enter. There will be many files in that folder, you're looking for the ones called com.apple.CharacterPaletteIM.plist and com.apple.CharacterPicker.plist. When you find them, you can probably just remove them, but it's always a good idea to back them up. Just move them to another folder or something.
When you've done that, reboot your computer, and see if the Special Characters window works again.
Best Answer
These are U+fb01 and fb02, Latin Small Ligature Fl and Fi. They are in Unicode really only because they were contained in legacy 8 bit character sets like MacRoman, but should no longer be used for anything. In modern technology such ligatures are created by fonts on the basis of the underlying codes for the separate characters, which is what should aways be input in place of option shift 5/6.