This Jayce build is for both Solo Top/Bottom Carry lanes.
It gives amazing sustain and gold farm during the laning phase.
It gives more than descent burst damage during team fights and late-game phase with high mobility.

If you like my build, please Rate it.

P.S.: More information and Screenshots will be added soon.

I decided to pick pure Attack Damage runes for maximum damage output.
Flat Marks, Seals and Glyphs and Level Quintessence.
All abilities except one (Lightning Field) scale up with Attack Damage and it makes it good choice picking such.
Also it is good for the Auto-Attack Burst Combo, which i will explain further in the Tricks chapter Heal keeps you alive when you are facing unexpected gang or fighting the opponent. Beware of the right moment to use it as many Champions have the ability to reduce Healing ( Katarina, Miss Fortune, Fizz, Tristana and etc.). Ignite can also reduce the Healing on you as well as the item Executioner's Calling.
Flash has both - defensive and offensive use. I prefer it over Ghost because Jayce has enough skills to reach certain target or to escape such. The possibility to pass through terrain is unmatched.
If you play Jayce as Solo Top you might replace Heal with Teleport.

I rely more on Q because i mostly stay inMercury Cannon stance to harass the enemies from a distance.
The second ability i max is E to give me more movement speed to chase/escape or down big part of the HP of my enemy, depending which stance i am in. To The Skies! / Shock Blast - Starting harassing ability.

In case your team makes invade or gang on level 1 or in any other fight use To The Skies! to come closer to a target and slow and damage him and his allies around it (if any). If he manages to run away from you and your team you can catch him by changing to Mercury Cannon stance and fire a Shock Blast. Thundering Blow / Acceleration Gate - Required for burst combo.

Use Acceleration Gate to open gangs or to prevent such.
Use Thundering Blow to burst an enemy. Lightning Field / Hyper Charge - Self Utility and brief AoE Damage

Lightning Field is needed to regain mana during the laning phase by hitting enemies.
Hyper Charge is amazing self ability for the middle and late game.

Starting with Boots and 3x Health Potion.
Rush Philosopher's Stone followed by Avarice Blade and Heart of Gold. These items will allow you to be more sustained while gathering gold in the same time. Keep them until you have to replace one for another item, when the inventory slots are full. (i.e. sell Avarice to get Infinity Edge)
The next goal is to build Trinity Force. If you overextend well you will probably build it up all in once, otherwise start with Sheen, then Phage and last Zeal. Then combine.
Upgrade the Boots to Mercury's Treads.
Buy Vampiric Scepter, B. F. Sword and upgrade to Bloodthirster.
Depending on your opposite team decide on which item is more needed and build it: Guardian Angel or Infinity Edge.
When you have bothIE and GA make the last item: Phantom Dancer. Why do i get these items.
gives Tenacity which is very helpful stat. It allows you to escape all the roots, stuns, slows, fears and etc. quickly and continue your move.
gives you Movement and Attack Speed that go well with Jayce; HP and MP to stay longer in battles; Chance to Slow enemies, which can proc from Shock Blast; Critical Strike chance; brief AD and AP and of course the 150% damage buff on next attack which helps you bursting enemies down.
Phantom Dnacer Give you enough Critical Strike chance which perfectly fits with Auto-Attack Burst Combo trick and additional Movement Speed.
make you more sustainable in fights.

Tricks
Reducing the Incoming Damage:
Whenever you have your ultimate ready and you are about to be attacked by a skillshot or other kind of ability change to Mercury Hammer stance to gain bonus Armor and Magic Resistance and lower the attackers damage. This is only if you are in Mercury Cannon stance.

Auto-Attack Burst Combo:
Using Hyper Charge will make your next 3 basic attacks to deal 130% damage instead of 100%. With the bonus Critical Damage from Infinity Edge and the masteries you will land 4 attacks in less than 2 seconds for real high amount of damage. Note that Hyper Charge resets your Attack Timer so you use immediately after you make 1 basic attack to optimize the damage.

Jungle Support:
When helping your jungler with the Golem or Lizard first enter Mercury Cannon stance and hit the mob when it is pulled. This way you will reduce its Armor/Magic Resistance with 10% so it can die quicker. Right after the hit change to Mercury Hammer stance and hit again, for the bonus magic damage.

Escaping Gangs:
When you are about to be ganged, follow these tips to stay alive.
If you are in Mercury Cannon stance use Acceleration Gate close in in front of you and instantly change to Mercury Hammer stance. The gate gives percentage movement speed, while the passive bonus from changing stances is flat. That way you will have highest possible movement speed you can gain from yourselves.
If you are in Mercury Hammer stance immediately change to Mercury Cannon stance and cast Acceleration Gate for the bonus movement speed.

Facechecks:
You can check bushes from safely place.
Use Shock Blast or Acceleration Gate. They both reveal Fog of War.
To reveal 2 bushes, like the ones on top/bottom lane, use Acceleration Gate in the first bush and fire a Shock Blast through it to reveal the second bush.

Laning Phase
Harass opponents with Acceleration Gate followed by Shock Blast.
Mana: Immediately after harassing move change to Mercury Hammer stance and autoattack minions to regain mana from Lightning Field plus the regeneration from Philosopher's Stone.
Health: Having Vampiric Scepter plus the regeneration bonus from Philosopher's Stone you will be able to refill quickly. If you are on bottom lane your supporting teammate will land you a hand.

Common Fight
Mercury Cannon stance > Acceleration Gate > Shock Blast through the gate > Hyper Charge and Auto-Attacks > pass through the gate and change to Mercury Hammer stance > To The Skies! > Lightning Field > Thundering Blow > Repeat