There are 5 rune cores per hero. The first one is unlocked for all heros. Unlocking more rune cores requires 4 different crystals. The crystals come from blessing your troops for sky city boss fight or achieving milestone damage amounts for each boss. The milestone damages needed are quite high. The most likely source of crystals will be from blessing troops because both gold and diamond blessing yield crystals.
Upgrading each rune core requires runes, gold, and sky city items. Grey runes will be most likely used because everyone has an abundance of grey runes. The bottle neck is sky city items. You can get them from sky city shop or other markets. Sky city items costs sky city rewards from boss fights, but you can buy them for gems from the other markets.
Your rune cores begin at low quality. When you unlock 3 rune cores and upgrade them to +3, you can advance all runes to mid quality (for that hero) for free. The next tier requires 4 rune cores at +6. If that trend continues (I will find out tonight), the next tier will require 5 rune cores at +9. Advancing the quality of rune cores increase the stat bonuses. Each rune core has 4 stats. Those stats are different for each rune core and each hero. It seems every stat is available in the rune core system (e.g., ap, ad, health, mr, ar, dodge, hit, crit, ar pen, magic pen).