The following is my process and the cards/upgrades I prioritize for high covenant Melting Remnant runs. In my opinion, Hellhorned is the strongest allied faction for Remnant due to high-damage units, imps with summon abilities that can be reformed, and armor/rage spells.
I reset the run at the start until Molded (reform a unit) is one of the starting cards and the Burn Bright (burnout 3) option is available for Rector Flicker. In my opinion, Burn Bright is the strongest champion upgrade and having a reform card mitigates and eventually negates the downside of burnout.
My early game priority is upgrading at least one copy of Molded with -1 cost and Holdover. Being able to guarantee the ability to reform units and your champion each turn (once you initially draw it) is crucial. Not having a reform card in hand the turn after your champion burns out can be devastating. Personally, I like to ideally have two upgraded copies by endgame.
In terms of boss upgrades, it really depends on the individual run, though I do tend to take at least one unit space upgrade. If you are allied with Hellhorned and end up with a Demon Fiend, you probably want a mana upgrade. Draw is especially valuable if you weren't able to get holdover on your reform cards.
When given the choice, I take Remnant units over Hellhorned/allied units early on. Specifically I am looking for Lady of the Reformed and Bounty Stalker.
Lady of the Reformed gives herself and other burnout units on her floor an additional burnout stack every time she takes damage. Using her as a tank for Rector Flicker and any other burnout units keeps them in the fight and is crucial against relentless bosses. I prioritize upgrading her health.
Bounty Stalker can become a huge source of damage if acquired early enough in the run. He gains 10 attack every time he dies, and with the ability to continually reform him throughout battles he can become a monster by the Seraph fight. I like to upgrade him with multistrike as a first priority and then either an additional multistrike or quick. A buffed Bounty Stalker is my primary target for card duplication. However, I tend to pass on him if he only shows up after Fel, since there isn't as much time to beef him up.
Other units that can be valuable during the run:
Lady of the House - 1-cost 25/40 size 2 unit with burnout 3. She does decent damage and can tank a bit on other floors.
Draff - 1-cost 10/5 size 1 unit with burnout 1 and multistrike. It does a fair amount of damage for only taking up 1 size spot. It is weak against sweep enemies unless it has upgraded health.
Paraffin Enforcer - 1-cost 15/15 size 2 unit with Strike: Friendly burnout units gain Rage 2. If you have an artifact that increases rage stacks or prevents rage from deteriorating, this is a solid pickup to line up with multistrike burnout units.
Paraffin Thug - 1-cost 20/3 size 2 unit with Slay: Gain 20 coins. Since store costs are increased at higher covenant levels, this can be a good option as a utility unit to build up wealth.
Formless Child - 2-cost 0/1 size 1 tomb unit with Extinguish: Return a defeated unit to your hand and enhance it with 40 attack. This unit gets better the thinner your deck is with units. You can use it to tank a hit and if you have a multistrike unit in your graveyard it's a huge payoff. The only downside is its high cost.
Spells that can be valuable in the run:
Wicked Blaze - 2-cost. Reform a unit and enhance with an additional 10 attack. A more expensive version of Molded with an additional benefit. I recommend upgrading with cost reduction.
Wicklash - 1-cost. Enhance a friendly unit with with +5 attack and apply Burnout 2. This is a good way to slightly buff burnout units while giving them a couple extra turns before they burn out. I wouldn't go out of my way for it but it is welcome in my deck if I get it.
Subsuming Blade - 2-cost. Deal 5 piercing damage with Slay: Increase damage by 3 permanently. Direct damage can be a lifesaver in taking out high-damage/low-hp units behind tanks. The piercing allows it to take out low-hp/high armor enemies as well as enemies with spell/damage shield. This is especially useful against Daedelus' mines when they have damage shield. I tend to upgrade with cost reduction and +10 magic power.
This is just my Melting Remnant playstyle and a number of others are viable, but this strategy carried me from covenants 20-25.