There are two separate currencies, and you shouldn't mix them up: class parts go to the bank and grow the meta tree, bio essence is spent at the terminal during the run.
Which Chapter Gives Which Part
Each chapter gives a single part type: System Core (sectors 1-5) prism, Scrap Foundry (6-10) alloy, Crystal Deep (11-15) flux, Organic Infestation (16-20) servo, Spacetime Rift (21-25) chrono, Omega (26-30) plate. If you want to grow a particular branch of the tree, you have to play the chapter that gives that part.
Where Bio Essence Comes From
Mine the green veins on the map: each tile gives +2 essence and drops a supply pack. Pick up the supply pack for +1 more essence and some health — if your health is already full, it turns into shield. Crates drop two supply packs. Essence isn't banked: if you don't spend it before the run ends, it's wasted.
The Terminal
The terminal puts 6 of its 10 offers on display, and each offer can be bought only once per run. An offer you buy leaves the display and nothing takes its place, so the display narrows on its own as the run goes on.
When to Spend
Costs go up with every visit — roughly 26% per visit. Since offers are one-time, waiting earns you nothing: buy early and you buy cheap. If you don't like the display you can reroll it, but within the same visit the reroll cost goes up with every roll (4, 8, 12…).
When there are no unshown offers left, the reroll button disappears. So "I've bought everything" is something you can see on screen — you won't waste rolls trying.