Legendary Byleth (M)

Weapon: Professorial Text

Assist: Reposition

Special: Sublime Heaven

Passive A: Swift Sparrow 3 / Life and Death / Other Atk/Spd-boosting skills 

Passive B: Windsweep / Watersweep

Passive C: Time's Pulse

Sacred Seal: Swift Sparrow / Atk/Spd Solo / Life and Death

• Byleth unapologetically takes many pages out of Young Merric's study notes running an eerily similar tome with Slaying and Null Follow-Up effects. With Windsweep added to Byleth's list of obviously plagiarized academia, he can run a deadly attacking build that simultaneously offsets his exploitable defenses as well. 
  • Thanks to the Null Follow-Up effect of Professorial Text, Byleth negates any follow-up hindrances associated with Windsweep. Thus, Byleth can initiate into many slower physical weapon-wielding enemies with absolutely zero fear of retaliation, allowing him to deal essentially free and massive damage in a short amount of turns.
  • You may also consider run Watersweep to disable the counterattacks of magic and dragon enemies instead, though the number of relevant enemies affected by Windsweep far exceeds that of Watersweep. 
• Since Windsweep requires Byleth to maintain a Speed advantage over his foe, it naturally follows that this build prefers high-tier Atk/Spd A and Sacred Seal skills. Swift Sparrow is therefore easy to recommend for both slots as it provides a straight Atk/Spd boost simply for absolutely zero drawbacks or conditions other than initiating combat. Life and Death drops Byleth's defenses, but that doesn't really matter on a build that blocks counterattacks anyway, and both its third and fourth tier variants slightly inch ahead of Swift Sparrow's second and third tier versions respectively. Other similar choices are up to your inheritance choices and discretion — Atk/Spd Ideal, for instance, technically provides the highest stat swing. 
• The combination of Professorial Text's CD-1 and Byleth's base Time's Pulse allows the instant full precharge of Sublime Heaven, allowing Byleth to deal a powerful burst of increased damage at the start of every turn. This is especially important to bypass any form of Special denial, such as Guard and especially Fallen Edelgard's Armored Wall.

Weapon: Professorial Text

Assist: Shove / Reposition

Special: Sublime Heaven

Passive A: Life and Death / Atk/Def Ideal / Miscellaneous Atk/Spd skills

Passive B: Windsweep / Lull Atk/Res 

Passive C: Joint Drive Spd / Distant Guard / Joint Drive Atk / Infantry Breath / Other "Within Two Spaces" skills

Sacred Seal: Distant Guard / Drive Atk / Tactics / Other Drive skills

• Seeing as Professorial Text features the ability to jointly share Null Follow-Up with fellow nearby allies, a build that's geared more towards providing additional support is definitely within Byleth's cards. Though a few skill slots are surrendered towards this goal, Byleth doesn't lose much of his offensive utility at all; he still utilizes Professorial Text and accompanying skills like Windsweep seriously well while balancing his buffing duties. Keep in mind, however, that this form of Null Follow-Up only applies if his teammate's Speed is greater than that of their enemy unit. (Translation: use fast carry allies only.)
• As usual of support units, Byleth favors supportive C and Sacred Seal slot skills to further enrichen his allies' combat. In particular, skills that provide buffs "within two spaces" are recommended to mesh in addition to the existing "Drive NFU" effect of Professorial Text while providing a less stressful positioning requirement overall. 
  • Joint Drive Skills make for a fine choice as they simultaneously improve a chosen stat between Byleth and his ally. Joint Drive Spd in particular synergizes directly with the Speed domination requirement of Professorial Text's NFU effect.
  • Distant Guard or Close Guard provides a more defensive stat swing.
  • You may also consider running infantry-specific skills such as Infantry Breath or Infantry Rush — though considering the infantry class's existing access to Null Follow-Up (meaning you can run pretty much any other battery unit) and the limited usability of these skills, they have less value on Byleth overall.
• Shove is a recommended Assist skill that is more specific to this build's playstyle than others. It pushes a chosen ally closer to a desired spot on the map while still abiding by his skills' two-space requirements. Else, other Assists are perfectly fine for your preferences. 
• The shifting goals of this build mean that while Windsweep + an Atk/Spd A-slot skill is endgame desirable, simply running his good base skills won't affect the main points of the build much on a budget. Hence, Atk/Def Ideal and Lull Atk/Res are listed while serving to improve Byleth's bulk and damage output as a nice bonus.

Weapon: Professorial Text

Assist: Any Assist

Special: Sublime Heaven / Blazing Wind

Passive A: Fury 4 / Life and Death / Swift Sparrow / Other Atk/Spd boosting skills

Passive B: Desperation / Special Spiral

Passive C: Time's Pulse / Joint Drive Atk

Sacred Seal: Life and Death / Swift Sparrow / Atk/Spd Solo

• This build is recommended more for general gameplay such as Abyssal map clearance — the previous two builds require basically no setup to function and are better built for PVP.  
• Professorial Text provides a subset of effects which basically opens Byleth's B slot for a couple of colorful options. In addition to some beefy additional stats +5 while in combat:
  • The Null Follow-Up segment of the weapon allows Byleth to basically run Desperation unhindered by skills that stop follow-up attacks, such as Wary Fighter or Twin-Crest Power. 
  • The fact that Professorial Text reduces Byleth's cooldown by -1 permanently means he can alternatively run Special Spiral with a desired AOE Special without requiring a Blade skill.
• For A-slots, most of the entire Atk/Spd pool is Byleth's oyster:
  • Fury is universally great for both mentioned strategies — the extra bulk and damage suffered after every combat means it's significantly safer to set up Desperation, and the visible Atk/Spd boosts alternatively optimize AOE Special damage (they take Byleth's visible Atk stat minus the enemy's Res for damage calculation. 
  • Life and Death provide larger visible Atk/Spd gains, but at the cost of Byleth's setup as it negatively impacts his defenses as well.
  • In-combat Atk/Spd swing skills such as Swift Sparrow or Atk/Spd Solo are also fair game, but are not as optimal for AOE damage.
• Time's Pulse is a fantastic default skill that's pretty difficult to replace as this build places significant emphasis on Byleth's offense and Specials. Generally, it precharges Sublime Heaven fully so that Byleth can start off with a bang at the start of any turn, and while it essentially turns useless after the first round of AOE, it makes it easier to charge up the first time around. 
• If you're concerned about optimal effect usage or simply improving damage, Joint Drive Atk or Joint Drive Spd are there as hybrid support / offense-boosting options. Notably, they just so happen to synergize wonderfully with the similar "within two spaces" Null Follow-Up effect from Byleth.