Can I ever play Undertale again normally?

If you're running the Steam version of the game, deleting files in the save directory won't be enough. Steam Cloud will automatically restore those files, and it also keeps a local cache of the files to restore them even if you're offline. While it's possible to clear these files out, it's complicated, and a simpler solution exists on Windows if you have sufficient access rights on your computer.

• Go to your computer's save location. (Default on Windows: [User folder]\AppData\Local\UNDERTALE)
• Right click the file system_information_962 or system_information_963, depending on if you are on post-genocide black screen, or post-genocide post-soul-sold.
• Go into properties, then security. Remove [Your username] and [administrators] rights to READ the file. Click apply.
• When you next start up the game, it will be as if you had never done a genocide run ever before.

If you're not playing the Steam version of the game, the simplest method to remove the consequences of completing a genocide run is to fully uninstall and reinstall the game.

Source: http://undertale.wikia.com/wiki/Genocide_Route#Consequence_Avoidance

If you find where the Undertale save files are located, you can delete them and start fresh. According to this Steam forum post, the save files are in appdata\local\UNDERTALE on Windows, ~/Library/Application Support/com.tobyfox.undertale/ on Mac. There is also a save manager available which one of my friends has used successfully, but since it's a third party program use it at your own risk: Undertale Save Manager.

Keep in mind that if you're playing Undertale through Steam, the Steam Cloud saves may attempt to overwrite your save file with the stored version when you try to remove it. Disable it to prevent this from happening.

The best way to clean everything up (including your Steam cloud) would be to use UGR (Undertale Genocide Remover). This is a third-party and, as usual, you should be careful with those. (I personally already used it and it worked fine with no apparent viruses.)

But don't think that YOU will have a clean conscience though. The app make sure to give you chills ;)

If not a steam downloaded version just go to ~/Library/Application Support/com.tobyfox.u and delete all files. Worked for me!