Don't stress out about this too much. The hardest one to get is the Leviathan core which comes from the Leviathan Labyrinth in the nexus hub. You get to the labyrinth by going west from the mission portal and jumping across the gap. The labyrinth involves a pretty annoying overhead platforming maze. There are a ton of videos on youtube that will show you the correct path.
For the rest of the cores you will get most of them from doing your 70,000 score in the arena.
You will also get a core from each of the game's bosses including Killington which is found behind the large trickster door in the nexus near Vulgrim. You open this door by opening all the trickster doors in the campaign. This will also require that you collect all 27 trickster keys.
As you play through the game, Vulgrim will sell some cores in his inventory. All the unique ones that he sells do not drop anywhere else in the game so you will need to buy at least the first rank of all of these.
He will also eventually unlock purchases that will give you random cores from different areas of the game. These random purchases will never be a core that you have maxed out. So you can keep purchasing these until it won't let you buy anymore which means you have rank 3 of all of the cores from that area. Personally I would wait to do this until after you have completed the campaign at least once AND have done the 70,000 score in the arena.
Once you have collected all the cores the achievement will pop. You can also browse through the campaign missions or the arena missions to see if the cores listed at the bottom of each map have all be acquired. They will be purple in color with a small wavy line through them if you don't have the core yet.
If you have all those cores then you are missing a boss core or the Leviathan core. You only need to complete the first second of the labyrinth and get the first Leviathan core for this achievement.