Lethality Jarvan takes advantage of many elements of his kit that the traditional bruiser playstyle does not. Compared to other assassins, lethality Jarvan is much more durable and offers hefty amounts of crowd control to match. Even with a damage oriented build, lethality Jarvan can still fulfill his role as a diver due to his W shield and items such as Eclipse. Where lethality Jarvan lacks in healing he makes up for with shields and damage, often mitigating equal or greater amounts of damage despite the lack of sustain items while still dealing consistent damage to match that of other assassins.

Lethality Jarvan has the potential to deal large amounts of damage through his natural kit and through items and runes to synergize. He has very high AD scalings on his abilities and is great at dealing with tanks and bruisers due to his passive and Q armor shred. The damage he deals can be applied on a target very quickly since his engage is tied to his damage, and most of his damage can be applied during his E+Q to ensure it is utilized at its fullest. Since knockups cannot be reduced through tenacity or QSS, his damage can be applied at its fullest potential as long as the target is caught in your dash.

Jarvan's ability rotation is pretty simple, and I am writing this under the assumption that you know what his rotation would typically look like. His kit is very simple but utilizing certain items and runes extends his potential damage output immensely. Runes and items especially have a major influence in his damage potential, specifically the Keystone Electrocute and the minor runes Sudden Impact and Shield Bash, with the items Eclipse, Black Cleaver and Prowler's Claw.

Electrocute is applied off of 3 separate abilities, and Jarvan can utilize this without even needing to expend 3 damaging abilities unlike with other champions. His E+Q+W combo procs it instantly, despite his W not applying any damage. Even if your E is missed you can still proc Electrocute by connecting a basic attack after dashing and shielding with your W while your target is airborne.

Sudden Impact adds additional Lethality and Magic Penetration after using a dash or blink. This can be used with Jarvan's E+Q, R, or by using Flash. This added lethality only applies after using a mobility spell or ability, so his first instance of E+Q or R will not benefit from it. It's damage output is best utilized in extended fights or when setting up combos for your R since it has the highest damage of his abilities and can easily hit multiple targets.

Shield Bash adds additional damage to your first Basic Attack after gaining a new shield proportional to the size of the shield gained. This can be applied through Jarvan's W or through items such as Eclipse, Maw, or Sterak's. Since Jarvan's W and the core item Eclipse both provide a shield that scales with bonus AD, the potential damage output of this rune is very high. Shield Bash has no cooldown but its damage is based on the size of your shield relative to any shields that are currently active. This means that if you are shielded and gain a shield smaller than the one you currently have, the rune will not proc. The damage is best utliized by activating your shields in order of smallest to largest and using a Basic Attack in between new shields. Jarvan's W will always be the larger shield out of it and Eclipse, so save procing Shield Bash off of it only after your Shield granting items have been utilized first.

Eclipse provides a burst of damage and a large shield after using two instances of abilities or attacks. This is easily utilized in his kit to provide the damage and shield instantaneously with his E+Q since his knockup is two unique abilities. It also has a low cooldown so it can be utilized outside of just his E+Q with any other form of damage from his other abilities or through his Passive and Auto Attacks.

Black Cleaver applies a stacking percentage-based armor reduction whenever physical damage is dealt to an enemy champion. This applies to any runes and items that deal physical damage. Since Jarvan is almost entirely physical damage, this item adds a lot of additional damage to his kit without needed any special requirements. His Q also provides a similar percentage-based armor reduction, for a total of 56% armor reduction with both appilied completely.

Prowler's Claw gives additional physical damage and a slow on your first Basic Attack after exiting a mobility spell or ability. This is easily applied in Jarvan's rotation since he has high Attack Speed and two dashes, activating off of his E+Q or R.

His rotation utilizing the aforementioned runes and items would be as follows:
P, E+Q+A, W+A, R

An auto attack procs Jarvan's Passive and Shield Bash from his Jungle item. E+Q Procs Eclipse and Sudden Impact, followed by an auto attack to proc Prowler's Claw, Electrocute, and Shield Bash from the Eclipse shield. W and an Auto Attack procs Shield Bash again and R after that finishes the rotation. At this point Black Cleaver and his Q armor shred should both be applied at their maximum potential assuming everything was hit in the rotation.
After the first initial rotation you can repeat the process without R. After the first totation Use your Passive on unique enemies with W and Shield Bash whenever possible and knock up enemies with E+Q when needed.
With a large mix of flat and percentage-based damage from the full rotation, the resistances and health of any given target should be irrelevant in your damage output.

Other items have synergy with his rotation but the most powerful ones for damage were mentioned above extensively. Other useful items can be bought situationally, with their potential damage being a compliment to their utility or sustain. These items are as follows:

Maw: Provides a very large magic shield under low health that can proc Shield Bash.
Ravenous Hydra: Provides AOE Physical damage that heals you to any attacks or abilities and stacks of bonus AD based on kills or assists scored without dying.
The Collector: Executes enemies at low health and gives 25% crit chance to any Auto Attacks or Passive procs.
Axiom Arc: Reduces R cooldown whenever a kill or assist is scored.
Death's Dance: Delays damage taken and removes the delayed damage and heals based on bonus AD after a kill or assist is scored.
Chempunk Chainsword: Applies Grevious Wounds to a target whenever you apply physical damage to them.
Edge of Night: Gives a regenerating Spell Shield that blocks the first ability hit.
Guardian Angel: Allows you to revive upon taking fatal damage at a high cooldown.
Spear of Shojin: Gives cooldown reduction to basic abilities scaling off of bonus AD.
Umbral Glaive: Frequently reveals nearby wards traps- Build if you take Zombie Ward.
Steraks' Gage: Gives a portion of base AD as bonus AD and provides a large shield at low health that can proc Shield Bash.
Serpent's Fang: Reduces incoming shields on targets whenever physical damage is applied to them.