Been trying scholar in depth 5 with various builds and here's what I think is the best way to build him based on my experience.
This build focused on every aspect of the game: early game, support and late-game DPS. I don't consider playstyles that require your teammate's help (like firepot) and heavy RNG-dependent setup. He's imo the most relic-dependent character in the game. I sort these relic effect by priority instead of just telling what is good so every one can make their own build with what they have.
Necessary:
* Continuous damage inflicted on targets threaded by ultimate art: Everyone rushes the nearest invader camp at the start of almost every D5 match. This effect makes invader camp much easier and speeds up clearing troll and banished knights castle significantly.
* Defeating enemies fills more of the Art gauge + 0 & +1: Need both +0 & +1 version so you can use your ult in every encounter. Scholar's remembrance relic is good because It has 2 best effects for him.
* Dormant power helps discover [ ]: While the rot thrusting sword is good, if that is all you can provide in late game, he falls short in term of DPS. He really needs to change his weapon pool for something better. I find katana, reaper to be the best options. Heavy thrusting sword and flail are good alternatives.
* Starting skill: He needs to make sure he's always useful with 0 RNG involved (sometimes you just can't access a merchant or get a bell from grave). A good starting skill is necessary. Hoarfrost stomp is the best because it's safe, area damage and pairs well with firepots for repeated frost-defrost. Blood blade is good too but too dangerous. Alternatively starting with wraith-calling bell is also solid.
Very high priority:
* Balancer relic: Scholar doesn't really have room for both so pick one. Red balancer is better for survivability.
* Allies targeted by character skill gain boosted attack: This turns the most niche character skill into top-tier damage booster skill like Ironeye's mark. It takes like one second to put damage buff on your teammate remotely (It's 20% physical damage and 15% affinity damage boost). Remember you need to scan at least one enemy to activate this. You can't directly buff your teammates with only teammates inside the scan.
High priority:
* Improved Endurance and Dexterity, Reduced Intelligence and Arcane: Applying ailments slightly slower is absolutely worth the damage boost.
* Attack power increased for each night invader defeated: With everything above, it's hard to fit invader relics into the build, but equip one if you have a relic this good.
* Improved affinity attack power, Improved physical attack power: I personally prefer affinity damage because most FP-consuming skill are either complete or partial affinity damage.
* Increase maximum HP, Improved physical damage negation, Improved affinity damage negation: For survivability.
Low priority:
* Increase maximum FP, Reduced FP consumption when using character skill on Self: FP management isn't that big an issue for scholar.
* Items confer effect to nearby allies: Really depends on your playstyle. I find this to be a debuff personally because upgrading consumable makes you fall behind your teammates and you can't buff them and use your ult if you're always late. Not to mention that the longer animation makes upgrading waste more time and can kill you sometimes.
With the build above, I can always pull my own weight immediately from the start of the match. With good RNG, I can be a good damage dealer with the right weapon and skill attack up passive. At worst, I can still buff my teammates with character skill to boost our damage.