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!
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.
It is not advisable to go on a killing spree while doing a Relic Hunter run. Most NPC's are bound to one quest or another and even the smallest quests can have a relic at the end, quests like "Deceiving the Herd" are therefor better avoided. Aside from Nazeem ofcourse, do whatever you want with him.
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).
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.
This quest requires you to travel to Raldbthar where it is very hard to avoid fighting Alain Dufont. Killing him then instead of during 'Mourning Never Comes' will result in Muiri's Ring being unobtainable.
Make absolute sure you get Helm of Yngol before you talk to Jarl Korir.
Siding with the Empire will make Maven Black-Briar Jarl of Riften. She will not give you the option to become Thane if you have completed the Skooma Quest The Raid prior to siding with the Empire. Neither will Jarl Korir be available for the abovementionned Helm of Winterhold quest.
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.
In short, just make sure to be Jarl of every hold before doing the main questline or the civil war.
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. It will also mess with the Jarls which causes numerous problems with their fetch quests.