Keeper

Repentance 1.7.9 on PS5.

What you start with

The mechanics that actually govern this character.

Coin Hearts instead of red health: 2 containers to start, 3 once you have beaten Hush as Keeper. Every coin you walk over is spent, one cent per missing Coin Heart, to heal. Stats are damage x1.20, Tears -1.9, Speed 0.9, Luck -2, plus an innate triple shot that is NOT The Inner Eye (so Inner Eye can still spawn).
He always takes exactly one full Coin Heart of damage and can never sit at half a heart. That kills The Polaroid and The Negative half-heart triggers, Whore of Babylon, Empty Vessel, The Wafer and Cancer. The Negative's flat +1 damage still applies in Repentance.
Every heart pickup becomes blue flies: half red = 1, red / blended / half soul / rotten = 2, soul = 3, black / double / bone = 4, eternal = 5. Hard cap is 3 Coin Hearts; only Greed's Gullet (up to 7), Birthright (+1) and the Mother's Kiss trinket (+1) raise it.
Repentance-only rules that change everything: Devil Deals are paid in coins, 15 cents for a one-red-heart deal and 30 cents for a two-heart deal, and taking a hit no longer reduces Devil/Angel Room chance for him. Health Down pills are also removed from his pill pool.

Treat loose pennies as parked health. Deliberately leave one or two coins on the floor in every cleared room and walk back to one after each hit instead of hoovering everything up.
His fire rate curve is brutal at the bottom. Below roughly -0.77 Tears added Tears barely move fire rate; the wiki's worked example has his first three Tears Up pills give +0.08, +0.10 and +0.12 fire rate while the fourth gives +0.47. Never skip small tears ups, they compound past the hump.
Unlock order matters more than build quality: beat Isaac first for Wooden Nickel, then Satan for Store Key, then Hush for the third Coin Heart plus a starting coin. Normal mode is fine, marks carry upward.
Use Wooden Nickel on entering a room rather than mid-fight, so the coin it drops is already sitting there as emergency healing when you get clipped.
Sacrifice Rooms are his best room in the game: leave coins scattered outside, step the spikes, walk out, heal on a penny, walk back in. With 3 Coin Hearts this is nearly free reward farming.
Curse Room entry costs a full Coin Heart. Only enter with a coin parked at the doorway, flight, or Holy Mantle.
Slot machines, Shell Games and Beggars are healing stations, not gambling. Money you feed them comes back as health.

Strong on this character specifically, not just strong.

Head of the Keeper: enemies drop coins when hit, which is continuous healing. Paired with Soy Milk the coin stream is effectively constant.
Pyromaniac: every explosion heals a full Coin Heart. With Kamikaze!, Dr. Fetus, Epic Fetus or Ipecac he stops dying.
Greed's Gullet: the only real health scaling he has, up to 7 Coin Hearts, and with Deep Pockets it can reach 12 Coin Hearts at 600 coins.
Swallowed Penny drops from urns, so bomb every urn in Chapter 1. It drops a coin when you are hit; in Repentance the coin is no longer guaranteed but it is still his best trinket, and Mom's Box doubles it.
Placenta: regenerates one Coin Heart per minute with literally no downside, because the half-heart drawbacks that normally offset it cannot happen to him.
20/20: in Repentance this gives him a large Tears Up with a Damage Down (like Odd Mushroom Thin) and fires 5 tears in an arc with no fire rate penalty. Cursed Eye gives 15 tears with no added tear delay.
Tech X and Tractor Beam both funnel his three shots onto one target; the Rib of Greed trinket (his Ultra Greed unlock) biases room clear drops toward coins instead of hearts, which for him means toward health.

Picks that are neutral or good elsewhere and bad here.

The Inner Eye and Mutant Spider: in Repentance they give him only 4 and 5 tears respectively with a large fire rate cut, a net DPS loss on top of an already awful tears stat.
Guppy's Paw: in Repentance removing his last Coin Heart kills him outright. Abaddon no longer kills him but drops him to a single Coin Heart.
Bum Friend, Bumbo and Dark Bum: they vacuum pennies off the floor, and those pennies are your health bar.
Maggy's Bow, Candy Heart, Soul Locket, The Jar, Immaculate Conception and Judas' Tongue do nothing for him; Alabaster Box can only be charged by items that hand you soul hearts directly, never by ground pickups.
Health Ups are filler once you are at 3 Coin Hearts, though they still top you up if damaged. Before Hush is beaten they are gold, because any Health Up grants the third container.

Verified against 1.7.9 on console — anything patched, PC-only or Repentance+ only is called out rather than listed.

Free Curse Room: park a coin immediately outside the Curse Room door, tap the doorway to take the entry damage, grab the coin during your invincibility frames, then walk in while still invincible. Net zero health.
9 Volt charge farm: in the room immediately before a Boss Room, and in trap rooms, the doors do not auto-open after clearing, so you can stand there cycling Wooden Nickel or any active indefinitely.
Carry a Wheel of Fortune card into Hush and Mega Satan. Neither attacks until you approach the centre, so you can drop a slot machine at the edge and farm healing coins mid-fight.
Moving Box can be loaded with several coins and unpacked inside a boss room as a pre-placed medkit.
Not on 1.7.9: the Afterbirth infinite money loop from Mom's Box plus Swallowed Penny was cut in Repentance (you get a stronger drop chance, not endless cash), and the Afterbirth co-op-baby trick for surviving a 2-red-heart Devil Deal is irrelevant because Repentance deals cost coins instead of hearts.

Keeper is a normal (non-tainted) character: 14 secrets, and all marks on Hard mode unlocks Sale Baby.
Three of his marks are direct upgrades to himself, so route them first: Isaac gives Wooden Nickel, Satan gives Store Key, and Hush gives 'Keeper now holds... A Penny!', which is both the starting coin and the permanent third Coin Heart.
The rest: Mom's Heart gives Super Greed Baby, ??? gives Deep Pockets, The Lamb gives Karma, Boss Rush gives Sticky Nickels, Mega Satan gives Noose Baby, Ultra Greed gives Rib of Greed, Ultra Greedier gives Eye of Greed, Delirium gives Crooked Penny, Mother gives Keeper's Sack, The Beast gives Keeper's Box.
Greed Mode is unusually friendly to him because coins spawn continuously during waves; leave a few on the ground each wave and he heals his way through Ultra Greed.

Open the tracker for the ordered run queue, or how to unlock characters if this one is still locked.