Per the book, at 1st level, Elves can buy mithril armor, even in Zelkor’s Ferry (Rasmus Pye somehow manages to get his hands on virtually anything adventurers ask for). This “basic” mithril armor has no additional cost and has no additional benefits, other than not being iron-based. Thus, mithril chain costs 150 GP, grants 5 AC, has a -5 check penalty, and -5’ speed penalty.
If you have the funds, you can buy higher tiers of mithril armor, each tier costing the full value of the item. So, first tier mithril chain costs double (300 GP), second tier triple (450 GP) and third quadruple (600 GP). Mithril armor has a number of tiers equal to half its check penalty divided by two rounded down, and so chain has 3 tiers.
Each tier grants a +1 revocation of the check penalty, meaning tier 1 mithril chain has a check penalty of only -4, not -5. Tier 2 gets you a -3 penalty, and tier 3 gets you down to a -2 check penalty at the cost of 600 GP.
At the highest tier for a given piece of armor, the speed penalty is reduced by 5’, so mithril scale, chain and banded become no-move-penalty, and plate has a 5’ penalty.
Mithril armor must be purchased at the desired tier; it cannot be upgraded, as the smithing involved is too advanced to try to bring a lower tier up to a higher tier, so save for what you really want.