Immortal Archon is a build that dishes out a lot of damage with storm and archon while making you almost impossible to kill due to improved phase shift.

Lv1:
Close call between Overload (psistorm damage) and Psi-Infusion (mana). I prefer the former as the build already lacks damage compared to other dps heroes like Valla.

Lv4:
Here there's no clear winner. I usually take healing ward as it's great for pushing with your team, but depending on the map/situation feel free to pick up any of them.

Lv7:
Khala's embrace is great, especially if you took Leeching Plasma at 4. But for this build we're going with Static Charge as it buffs up our damage and is amazing for clearing neutrals.

Lv10:
Archon period end of sentence.

Lv13:
Prescience is mediocre, but we take it because of synergy with Dimensional Warp.

Lv16:
Dimensional Warp. This talent makes you pretty much immortal. Usually in a team fight you can use Dimensional Shift twice and get another one from Prescience.

Lv20:
Twilight Archon. This turns you into a powerhouse.

Tassadar is a pretty strong laner. You can use oracle to spot ganks or find out if the enemies are doing a camp next to your lane. You can push the lane with storm but only do so if needed (i.e. you need to help your teammates nearby).

Try to stay on your lane to soak up exp and only go away to help allies nearby, or to help your team push/take an objective. At early levels lane exp is so important that successful ganks may not pay off (exp-wise) if they take too long.

In teamfights you want to stay in the back and use storm and shield. If you're low on mana, prioritize shield.

Map objectives and enemy forts are very important now. Only lane when there's nothing else to do. You should try to join all teamfights.

You can start playing more aggressively in teamfights. Don't initiate, but pop your archon when it begins and don't be afraid to stay next to enemies. While in archon try to hit cluster of enemies (since your attacks splash) and switch targets if they run out of your range. Avoid chasing while in archon.

The most important change in the lategame is that death timers are now very long. Regardless, you are now immortal. You are the perfect bait for starting a teamfight - run into the enemies, try to get them to attack/use spells on you. Then shift and run to your teammates. Pop archon and kill them. Sometimes the enemies will keep trying to focus you. Even if you drop low, the autoshift will activate and heal you. If you're low on hp, pop shift for a strong heal. But only do it in archon if absolutely needed (or you lose potential damage).

Between your phase shifts and plasma shields you're incredibly hard to kill. Your only weakness is that your damage becomes low after archon ends. The enemies may try to get you to waste it, so be careful with when you activate it (make sure they can't just run away).