I didn't explore Blackreach much during the Elder Knowledge quest, and I also didn't complete the A Return To Your Roots quest. It took quite a while to fight through Alftand to get there the first time so I'm hoping there is a quicker way to get in?
Alftand is the only way in... initially. As you adventure, you can take elevators back up to the surface, from which you can unlock gates leading out into skyrim proper.
As per the UESP, here are the potential entrances into Blackreach:
While several of these locations are fast-travelable-to, the Tower of Mzark is likely to be the quickest way into Blackreach, as it's associated "wing" is quite small, with no enemies to boot.
By contrast, some of the other ruins, like Alftand, or Mzinchaleft, still have goodly portions of dwemer dungeon ruin between their elevator and blackreach, which will restock their dwarven spheres and spiders from time to time. Since Mzark doesn't have enemies to begin with, it will always be enemy free.
The Great Lift at Mzinchaleft (it's a separate location from Mzinchaleft, to the south) takes you directly into Blackreach, so is definitely the fastest way in.