Doing this achievement in Permadeath will also unlock the 'Take a Deep Breath' achievement, which is getting to the center on survival.
From the tutorial start of the game, once you've learned to create your warp drives and are able to warp to other systems, follow the primary quest line until you unlock the Anomaly primary quest line.
Once you've unlocked that, scan in space to have the Anomaly show up and go inside. Talk to Priest Nada and when he gives you 3 choices about the atlas path, the shortcut or helping with exploration, choose the very first one for the atlas path. He will give you coordinates to an Atlas station in another galaxy.
Now you want to start to follow the very first quest line for the Atlas path.
You will need a lot of warp cells. Recommend to try getting 3 hyper drive (s) modules to extend the warp cells out. Also a Class A explorer ship or even one with just a good hyper drive range. The modules can be bought on the space station for around 450 nanites each. From my experience I didn't even warp up to 200 times. In total I warped 142 times. Around 200k light years from the center, I had broken black holes while doing this. So depending on how far you start and if they patch the black holes or not, I would make 150-200 to be safe, while using the 3 (s) modules and an explorer class A.
Beating the Atlas path gaining access to seeing the black holes at anytime and then just keep jumping from black hole to black hole until you get to the center. Make sure your hyper drive is completely full once you're at the end to be able to jump to the center.
Don't worry about the distance from the last system to the center.
Note: Once someone is at the center, their friends can join them and warp to the center to be able to unlock the achievement for themselves.
Also you can save on a planet or the space station in the last system, warp to the center and as you do that, you can close your game and the reload your save to stay at the end with your achievement still popping.