Just using the Hidden Carina Nebula texture pack isn't enough, you need to find the texture and use it to replace the end sky texture using the instructions near the bottom of the page.
Versions 1.2+ are compatible with optifine! (using optiforge).
Version 1.1 doesn't work with optifine, so if you want to use it with optifine, you need to get either version 1.2 or 1.0 of the mod. You can find version 1.0 under the "files" tab above. Also, there are different instruction on how to prepare the resource pack for version 1.0.
For version 1.3, I found that minecraft needs to be launched with the resource pack on, and the resource pack can't be taken off during gameplay, or reapplying it will result in a broken end sky texture.
Version 1.2 of this mod also works with sodium, which is an amazing fps increasing mod. However, version 1.3 didn't seem to work with the settings I had.
Forge verion here.
Use with a resource pack!
I made a resource pack here that uses the custom sky without having to edit the resource pack:
For versions 1.2 and 1.1:
Unless you are using my resource pack that I made, you need the file assets/minecraft/textures/environment/custom_end_sky.png to be a custom sky image from a custom sky resource pack, which you can find in the assets/minecraft/optifine/sky/world0 folder. It isn't necessary, but if you don't add the file, the end could look strange.
For version 1.0, the file needs to be named end_sky.png instead of custom_end_sky.png
When using version 1.0, it could make the end sky, look weird if you either use only the resource pack, or only the mod. I recommend using the mod with the resource pack, though no major bugs/glitches would occur if you only use one or the other.
Also, if you've ever wanted Minecraft discs to play different songs, check out my squares album resource pack! It changes all of the minecraft discs (other than pigstep) to play songs from the squares album.