Akkala Tower is in northeast Hyrule. This page has advice for finding and climbing it.
Location of Akkala Tower
To reach Akkala Tower, warp to Lanayru Tower, then go to Soh Kofi Shrine, which is just north of Lanayru Tower. From there, go north over the hill to reach the path. Go northeast along the path. When the path splits, go right (or you can go left to find a stable, then after you take a horse, go southeast to return to the fork in the road, and go left).
There will be another fork in the road. Go left and you will soon go across a bridge. As soon as you are on the other side of the bridge, start looking to the right until you see some stone steps. Go up those steps, then turn left so you are going north. Go up this slope and you will reach the steps that spiral up the hill to the base of the tower. There are Guardian Skywatchers patrolling the area, so be careful to avoid them.
Climbing Akkala Tower
When you reach the top of the Akkala Citadel Ruins area, you find that it is covered in Malice Goo. The north side of the area has a small set of stairs. Go up them and you will see a Glowing Eyeball through a hole in the gate. Shoot the eyeball and a thin strip of metal will fall. The door is too far to reach with the metal, but if you go to the left, you can set the metal down in the gap between the two sections of red carpet. Walk across it to reach the other side.
You can see the base of the tower here, but you can't safely climb it from this spot. Use Magnesis to pick up the thin piece of metal again. Look for the ledge to the left of the tower that has Malice Goo along the edge. Use the thin piece of metal to make a ramp up to that ledge. You still can't climb the tower from this location. Look down to see an area below you that you can land on. Float there, then climb up to the ledge where you can see a Bokoblin. Kill it. From here you can see a broken stone archway near the tower. That archway is your destination. Once you reach it, carefully climb it so that you don't touch the Malice Goo. From there, you can float to the tower and climb the rest of the way up.