This isn't a guide. This is an analysis. I don't assume you're new to Jax or League, so I'm going to be straight into analysis of the why, how and when's of these matches.
Game one is first, I'll update the rest of the semifinal games, including the game Impact denied Jax on Expession. I'm also working on the finals games with SKT, although they were virtually identical in all three games.

Expession became an absolute monster in this game. He opted for mainly penetrative stats in early game with his rune set and masteries which worked in his favor as he rushed hybrid damage with the ever popular recently buffed Trinity force.

This choice was made for two ressons, first - his lane opponent.

Jax is a difficult opponent to face in lane. Because his first few levels are rough, because he relies so heavily on abilities to close the distance and deal the brunt of his damage, he opted for a crystalline flask and health pots first buy. We saw Impact for SKT taking advantage of this, putting two levels into his q and e, ignoring the shield for level 3 and punishing Expession when he tried to farm. By the time Jax hit level level 4, he had the combo on level 2 empower and was able to bully out Shen who gave up trying to poke him out and used his Q for virtually exclusive farming. An early back for Dorans Blade and health pots put shen ahead slightly until level 4, however. Jax's E is able to stop Shen's Stand United, with Leap Strike for easy gap closer, so he was a very strong pick.

Second, knowing SKT would want to shut down the renown Cain and PRaY in bot lane, they needed to deal damage out elsewhere. This choice of a carry top lane was because the knowledge that Bengi would be focusing bot lane.

Runes And Masteries -
Against a tanky opponent in lane, penetration always gives better value than flat damage. Shen is a very, very strong damage-soak with a shield and sustain so being able to bypass those resistances is a must. That's why we see Expession running the hybrid damage and penetration in both runes and masteries, bringing the early bonus from Weapon Expertise and Arcane Knowledge in the offense tree. Taking Summoners Wrath gave him both Attack Damage and Ability Power on the use of his Ignite, something which hybrid champions benefit from well and common to see on champions like Orianna, Ezreal and Corki, Jax is no exception.
Although he only gained a small amount of AP from his masteries and triforce, the bonus damage from Spellsword was quite decent when his team had Baron, which they prioritised due to the Corki ADC. Basic top lane 9 points into the defence tree, although important to note the respect points into Magic Defence over Armor for that early game poke from Shen.

Runes were interesting, but not uncommon - Maxing hybrid damage through penetration to ensure his base damages would eat through the enemy defense. With one Quint of hybrid pen, 9 marks and two glyphs of Magic Pen, espession has 9.9 armor pen and 8.2 magic pen.
Combine that with the 5 from Sunder, the 8% each from Weapon Expertise and Arcane Knowledge that means an opponent with 100 armor and magic resist is effectively on 72.5 armor (100-(100/8))-(9.9+5) and 74.3 MR. The flat penetration being much more effective in the early game.
We see him taking flat armor, a common choice, and flat magic resist, probably a direct answer to the early game poke from Shen. The 4% lifesteal rounds out his runes giving him that little bit of early sustain while triforce is being built. Interestingly though, the bonus damage from Empower, which he maxed does not apply lifesteal. It's likely that he wanted to maximise the sheen and triforce damage proc so opted for this over leap strike, which would have been the obvious choice for an ad heavy Jax build.

Items -

Triforce buff has been seen in play across the championship series and in almost all levels of play. Virtually every champion who can benefit from the passive Furor and damage proc is able to viably get this item, however AD focused champions seem to get the most utility from it's passive and overall stat bonus. Passive damage from sheen procs on bonus AD, movement speed on hit and balanced all around stats make is an amazing item for Jax, especially if he is able to complete it before the enemy can answer with a high level item like Sunfire Cape or Randuins Omen. Expession rushed this even before boots, setting his 14 minute build at Dorans blade, triforce and crystal flask.
Maxing empower first enabled him to combo Leap Strike, basic attack with triforce proc, basic, reset basic with second triforce proc and damage proc from Grandmasters Might, which saw him absolutely demolishing the health of Impact's Shen in lane. By this stage he had pushed through first turret and had to roam but was yet to buy boots. We see him returning to buy a Bilgewater Cutlass rather than the more obvious boots - the active of cutlass is used post-leap strike to ensure that he can keep them in range for auto attacks and damage procs, we see him use this on Faker's Ahri in a 4 for 1 teamfight in mid in favor of NJS.
Off the back of successful teamfights, the global gold from towers and he backed to get both Merc Treads and BotRK, merc treads for obvious tenacity against the Ahri charm since he focused the backline and BotRK for health shred and the obvious damage spike that comes with it.