This is my go to build for [Lavos] when I just want to get shit done. Due to having 3 Umbral mods it is a bit high investment, but there are variants of similar strength that don't use umbral mods.
You can invest an additional Umbral Forma to save 2 regular forma (Not needing Polarization for one of the augments and [Cunning Drift]). This gives you a bit more flexibility with alternate builds. You can also go for 3 Umbral Forma, which saves another regular forma (From the other Augment or [Archon Continuity]).

Shards

[Lavos] scales off of basically every stat and doesn't have particular breakpoints, meaning most shard choices are good. You can build with green shards for full armor strip or Toxin scaling, Red shards for Strength/Duration, Violet for primary weapon damage, etc. Regardless of what you go for I would recommend at least 1 Topaz shards for casting speed, as [Lavos]'es 4th ability, Catalyze, has a fairly long animation. Ofc avoid any energy buffs from shards, as [Lavos] doesn't use energy.

For the above build I run 2 casting speed shards and 3 strength shards. Tau and regular both work, so just go with whatever you have on hand.

Arcanes

[Lavos] loves strength for maximizing damage with his 4th ability, Catalyze, so strength arcanes work great. Molt Augmented and Arcane Bellicose (which gives +53% strength with [Lavos Prime]'s 800 base health, and [Umbral Vitality]'s +180% health bonus) are great picks. I like running both, but swapping out one of the two (Molt on Umbral Builds and Bellicose on non Umbral builds) also works well.

[Lavos] can also struggle a bit for healing if you mess up your ability rotation or don't have enough enemies in range, so a source of healing like Arcane Grace or a source of armor like Arcane [Guardian] work great for bolstering survival. Though a companion with [Hunter Recovery] also covers this.

Arcane Impetus is also a good choice, as [Lavos] can potentially apply every elemental status for massive bonuses. Though 5 statuses is more realistic giving it a respectable 30% strength bonus and 15% efficiency bonus. Its a good arcane to slot in for missions where you wont be able to build stacks for Molt Augmented, or for players who constantly change elements with every cast.

Finally any arcane that buffs your primary weapons work great, especially if you want to lean into being a weapons platform. Arcane Hot Shot is great, as [Lavos] can apply heat with every ability.

Mods

I go with full Umbral mods as [Lavos]'es high base HP and armor make him an excellent health tank. I run both augments for a generalist build that can both be an ability casting and weapons platform giving a ton of flexibility. And since [Lavos] scales off of all 4 of the basic stats I don't use any corrupted mods.

Aura

For the aura I run [Corrosive Projection] simply because its a solid mod that fits the existing polarity. If you want to spend an additional forma feel free to slot in [Growing Power] instead. For a Mecha setup use [Mecha Empowered].

Exilus

I run [Cunning Drift] for a bit of extra speed and to round out ability range to a clean 160%, which gives his 4th ability, Catalyze, a very respectable 40m range. You can also slot in [Power Drift] or [Speed Drift]. Never run [Primed Sure Footed] on [Lavos], as his passive gives him complete status immunity, which includes almost every knockdown effect.

Survivability

To help with health tank I run [Adaptation], which is on average a 70% damage resistance (Considering normal damage distribution of enemy attacks). Other mods that give additional layers of protection also work wonderfully, [Rolling Guard] being another great mod to use. If you are running the Mecha set this is the mod your would swap out for [Mecha Pulse].

Non Unbral

You might not have the Umbral Mods leveled up, or you might not have an Umbral Forma to spend. In either case you can swap to the following builds. Giving you a similar level of strength while swapping out [Umbral Intensify] + [Umbral Fiber] + [Umbral Vitality].

[Precision Intensify] + [Health Conversion] + [Archon Vitality]
For a full investment build that uses no Umbral mods this is a great choice. [Health Conversion] can be swapped for [Mecha Pulse] on a non umbral Mecha build, or Steel Fiber for more consistent tank.
If you aren't running the Umbral set then alternate healing/defense from arcanes becomes more valuable, so I would also slot in Arcane Grace / Arcane [Guardian] over Arcane Bellicose in this build.
This build is lower investment then the full Umbral build, requiring 3 forma.

[Umbral Intensify] + [Umbral Fiber] + [Archon Vitality]
This is a hybrid build. Making use of 2 Umbral mods to gain a bit more armor and Strength without requiring Umbral Forma. An alternate version of this is running [Umbral Vitality] + [Umbral Fiber] + [Precision Intensify]. This gives higher stats, but loses out on a universal strength buff and [Archon Vitality]'s heat status doubling.
This build Is higher investment then a non Umbral build, requiring 5 forma.

[Precision Intensify] + [Steel Fiber] + [Vitality]
This is a transition build, something to slot in when you want to eventually use a full Umbral build, but don't yet have Umbral forma.
This build requires 4 forma.

Mecha

This is the Mecha version of the build, making use of the mod set to massively increase armor while also spreading tons of status to every enemy nearby.
For any Mecha build you need both [Mecha Pulse] and [Mecha Empowered], so all Mecha builds swap [Corrosive Projection] for [Mecha Empowered].
For a full Umbral Setup swap [Adaptation] for [Mecha Pulse]. This costs an additional forma.
For a partial Umbral Setup swap [Umbral Fiber] for [Mecha Pulse].
For a non Umbral build swap [Umbral Intensify] + [Umbral Fiber] + [Umbral Vitality] for [Precision Intensify] + [Mecha Pulse] + [Archon Vitality].
For the partial and non Umbral builds you can swap [Adaptation] for [Steel Fiber] for more consistent bulk.

Make sure to pair with a Kubrow running the other Mecha mods.

Augments

While I love running both augments, if you want to lean into a casting or weapons platform build you can swap them out for other mods. On a casting build swapping [Valence Formation] for [Streamline] gives you better cooldown reduction for fasting ability cycling. Alternatively swapping out Swift Bite for a mod that supports your chosen weapon/Helmith ability works great.

For any build, swapping an augment for a mod with a Naramon polarity will save you 1 forma. Streamline, Constitution, Natural Talent or an Augur mod works well.

Helmith

[Lavos] has a very solid kit that synergizes well with itself, making other abilities unnecessary. Especially since removing a base ability also removes his ability to apply one of the base elements. If you do choose to use a different ability, the first pick for removal is, in my opinion, Catalyze. This may sound strange since most builds remove Vial [Rush], but personally I feel if you want to slot in a new ability, its best to focus less on dealing massive damage with his 4, and instead focusing on supporting weapons or that abilities niche while maintaining utility.

Ophidian Bite is your healing ability, Vial Rush is CC and Movement, and Transmutation Probe is team support for energy/ammo economy, removign any of these abilities massively decreases [Lavos]'es utility over other frames. Catalyze on the other hand is a long cooldown nuke that requires a lot of priming to be effective, making it the clear weak link.

Additionally removing heat has the lowest cost. Toxin, Cold, and Electricity have Viral, Corrosive, and Magnetic, three status effects that are incredible for weapon support, letting you build for full DPS while still being able to support your damage with statuses.

If you do remove Catalyze on partial/non Umbral setup swap [Archon Vitality] for regular [Vitality], as you no longer apply heat with abilities.

Final Notes

Overall [Lavos] is a very flexible frame who doesn't require any particular stats, but scales off of basically all of them. Giving you a lot of option for changing his builds to suit your personal tastes. [Lavos] has solid Bulk and good healing letting him health tank effectively in normal SP missions. He does start to struggle in endurance missions, in which case using outside sources of invulnerability becomes necessary.

Also it would be a crime to not mention his 5th ability, the [Cedo]. [Cedo] being [Lavos]'es signature weapon synergizes perfectly with his abilities. Here's the [Cedo Prime] build I would recommend to pair with this build:

Use [Valence Formation] to imbue the weapon with Viral and cast Ophidian [Bite] to proc [Archon Continuity], and you can apply all of the following statuses: Blast, Gas, Radiation, Magnetic, Viral, Corrosive, Heat, Cold, Electricity, and Toxin. For 10 total statuses.

Giving the [Cedo] a 1400% damage bonus (10 x (80% + 60%) )

Giving Catalyze 2^10 damage. With max Molt Augmented stacks (60%), 3 Tauforged Strength Shards (45%), [Umbral Intensify] (77%), and Arcane Bellicose (53%) you have 325% strength. Making Catalyze deal 4000 x 2^10 x 3.25 = 13,312,000 Damage. Not including bonuses from Viral/Magnetic.
If we remove Corrosive and Blast, as one only applies directly in front of [Lavos], and the other has a very short duration, we have 8 statuses, for a more realistic 3,328,000 damage, again not including Viral/Magnetic bonuses.