Aim to explore and discover lost places, artifacts and history, Build the guild house from the ground up as the guild master, then recruit new members, each with their own unique back stories and special guild functions. Explore ruins and excavate relic fragments to re-assemble into artifacts to display in the museum. Set up a guild store with the quartermaster and much more.
As a Relic Hunter you want to "collect 'em all". Finding over 2200 items scattered all across Skyrim is not something that happens overnight. Consider these major tips when adventuring across the wastelands of Skyrim, Wyrmstooth, Falskaar, Hammerfell, Cyrodill,...
Auryen can wait, look at those books man!
The road from the Solitude docks to the Dragonborn Hall is filled with unprotected stuff like books and gems. All of these things belong in the museum and will help get you to those important numbers 250 and 270.
Useless junk is never truly useless.
Making replicas will be an important factor as soon as you start doing quests. Most of these replicas require fancy crafting materials that aren't used at all in Vanilla. You will need things like Human Skull, Linen Wrap, Quill, Roll of Paper, etc more often than not.
Carry weight is the key.
Bandolier is good mod to help solve this major problem a Relic Hunter will face. Getting your carry weight to 700+ as soon as possible will help you a long way ahead. Other mods like EFF or AFT will help you too. Don't hesitate to hire an army of pack mules to follow you around.
Give that back? Hmm, maybe next year.
Most quest items are weightless. You can use this to your advantage by not giving it back until you really no longer need it. Examples are Ancient Nordic Pickaxe, Balgruuf's Greatsword, etc.
Boots of Springheel Jack
Right Eye of Coldharbour
Sword of Ancient Tongues
Getting your archeology skill up is a major part of a relic hunter run. As soon as you get your Explorer's Society guild house, you have acces to the special skill trees. Look out for excavation dig site's in every dungeon, they yield important fragments for your Hall of Lost Empires. The amount of fragments you need for every craftable items are listed below. Note that these numbers would require over 250 minings to get (without the perks).
Skills, perks and effects
Prowler's Profit (No Stone Unturned)
Although the quest is a real pain, the reward is a must-have. This effect will allow you to find 3 gems in every single container. That means every single burial urn will contain treasures for you to display or sell.
Extra Pockets (Pickpocket perk)
Exhume/Scavanger/Salvager (Academia perks)
Pay extra attention when entering these places because the relics here will forever be lost if they aren't found on your first (and last) visit.
Nord Head (alternate esp available on trophy head mod page that places the head on throath of the world)
Dragon Priest Staff
Cloak of Nahkriin
Left Eye of the Falmer
Right Eye of the Falmer
Cloak of Morokei
If either of the masks Morokei or Nahkriin are missed Kohnarik becomes unavailable. Without Kohnarik most items from Konahrik's Accoutrements become unobtainable.
Ruby Paragon -> Auriel's Shield
Circlet of Waterbreathing
Bloated Man's Grotto (before I'll Met By Moonlight)
Bolar's Oathblade (before I'll Met By Moonlight)
Thalmor Trophy Head
Any Orc Stronghold except Largashbur
The Forgemaster's Fingers (Not obtainable if you are an Orc or Blood-Kin)
Ancient Shrouded Armor (not obtainable if decided to destroy the Brotherhood or neglect to kill Giaus Maro in public)
Quests to postpone
Siding with the Empire will make Maven Black-Briar Jarl of Riften. She will not give you the option to become Thane.
Siding with the Stormcloacks will make Thongvor Jarl of Markarth. He will not give you the option to collect Hrolfdir's Shield.
Finishing this quest will mess with the guards in some of the holds, making their outfits unobtainable.
Siding with the Empire and completing the Civil War questline will cause "Blood on the Ice" to be unavailable, so the Necromancer's Amulet can not be recovered.
Dawnguard vs Volkihar Clan
Not sure what sides hold the most relics for you to display? They're listed below.
Amulet of Bats
Amulet of the Gargoyle
Ancient Vampire Leg
Ancient Vampire Ribcage
Ancient Vampire Hands
Ancient Vampire Arm
Ancient Vampire Head
Ring of the Beast
Ring of the Erudite
Bloodstone Chalice (need confirmation, I found it in the boss chest after defeating Harkon (as a vampire) so I assume Dawnguard find it there too?)