In contrast to all previous main quests, this one is the first one which is not activated automatically. This means that if you want to start it, you have to go to the marker on your own - it is located in Seattle Central district. There is enemy base located near the starting point of the quest, so be careful if you didn't destroy it yet. Move to the marker and accept starting new quest.
In the first stage of the quest you have to climb to the top of the Space Needle tower. Contrary to what you may think, it won't be difficult, although getting to the top will take some time, even with help of special powers. Start with climbing up the nearby container and then use the yellow ladder. After getting to the higher shelf use Smoke Dash to "jump in" to the nearby shaft. Wait until shaft "catapults" Delsin and catch metal objects to continue climbing.
After reaching larger platform located higher, locate another ventilation shaft and use it as a "catapult" (Smoke Dash). This time you have to aim Delsin during the flight towards yellow ladder shown on the above screen. After you catch the ladder, start climbing up. Carefully walk over the metal balcony and find a place where you can climb higher to the platform. In this place you have to use another yellow ladder.
When climbing up the ladder look for big bright wire located on the left and jump towards it. Resume climbing and do not get frightened by an elevator that suddenly starts to get down, because it won't collide with you. Keep climbing until you get to the small horizontal bar extending from the main construction. Rotate your camera so you can see a small balcony located on the left and jump towards it. Now you can start more traditional climbing, catching and jumping towards subsequent interactive edges.
You'll end up the climbing in a new small balcony. Turn you character towards a ladder visible in a distance, perform a jump and while you are in the air hold the X button to start the flight. In such a way you have to get to the ladder and use it to get to higher level. After climbing up the ladder catch a wire located over Delsin and use it to get to the tiny balcony (you have to jump at the end).
Jump into the shaft which will allow you to get to the main tower observation deck. Eliminate all D.U.P. soldiers along your way - it should be a piece of cake. Eventually you have to locate a red ventilation shaft leading to the large tower rooftop. Watch a short cutscene showing an anchor which is your new target and prepare to fight much more difficult battle.
Right after fight beginning start running over the rooftop, avoiding mainly the enemy with powers, who was shown on the previous cutscene. Avoid his distant attacks (one of them can stop you in place) and try to counterattack from a distance. Once you eliminate the elite opponent, deal with common soldiers, avoiding their fire and grenades. Do not forget about hiding behind metal covers when Delsin get seriously injured and locating smoke holes when you running out of smoke. In addition you can look around for generators and use them to create small explosions.
After the fights end, approach the anchor and try to destroy it, thanks to which you will learn that you need to unlock a new power first. Find a metal housing around, which contains a Core Relay. Destroy the housing, thanks to which Delsin obtain power named Orbital Drop. This attack requires you to charge a separate bar, before you can use it (the top-left corner of the screen) so, run up to the nearby enemies and stun them (if you play as the hero) or finish them off (if you play as the villain).
Once you sufficiently charge Orbital Drop, press Down on the D-Pad to activate it. The game will allow you to control Delsin's flight (left analog) but you do not have to hit the anchor directly, because the radius of this power is HUGE. At the end of the quest you'll have to choose which flag will be hung on the Space Needle tower (your choice is irrelevant). As a reward you will unlock main quest Chasing the Light.