I had the same problem upgrading to iTunes 12.3, but with the added wrinkle that I couldn't repair Apple Software Update because the cached copy of AppleSoftwareUpdate.msi was missing. It apparently got deleted by some cleanup program. In case you also run into this problem, you will have to get a new copy of AppleSoftwareUpdate.msi from the version you have installed to run the repair. THE 12.3 VERSION OF AppleSoftwareUpdate.msi WILL NOT WORK. (No, really. I tried that. It was not fun.)
1) Find a copy of your version on one of the old software archive sites. If you upgrade regularly that should be 12.2.2. Make sure you download the right bit size for your OS, 32 or 64 bit.
2) Open the installer file in your favorite ZIP program. If you don't have one, get one. I like 7-zip since it is free and has no adware or spyware. DO NOT run the installer; run your ZIP program and open the installer from within the ZIP program.
3) From inside your ZIP program, extract the "AppleSoftwareUpdate.msi" file from the installer file, and remember which folder you extract it to.
4) Go back and try to repair the Apple Software Update again. When it tells you it can't find AppleSoftwareUpdate.msi, hit "Browse" and navigate to the folder where you extracted it in step 3 and then proceed through the rest of the repair.
5) You should now be able to install iTunes 12.3.