Mount Riley is the tallest point on the Chilkat Peninsula just south of Haines.
Arguably Mount Riley might not be part of the Takshanuk Mountains seeing as there is a lull between it and them which the town of Haines resides in, but I think what formed them formed it and the Chilkat Peninsula. There are a few ways to approach Mount Riley.
The most common route starts at about 3 mile Mud Bay Road. There a large pull out and easily spotted trail head. From this approach it's a almost 3 mile hike to the summit. If you decide to start this way you will start on a mostly level approach through some boggy areas with streams and board walks. Then almost immediately once you're past there you will begin a fairly steep climb, switch backing up. When you get to the top of that part you will join the Lily Lake route. Go right to continue to the summit of Mt. Riley.
Another approach is the fore-mentioned Lily Lake trail. Go to the top left of Historic Fort Seward and head down FAA Road. Turn right before the Haines city dump and park and begin walking up the town water access route. Continue on that and you will meet up with the Mud Bay route. Go straight to summit, right if you want to go out to Mud Bay Road.
Another way to access the summit or leave if you approached from either of the previous two routes is Portage Cove. To get to the trail head, head south out of Haines on Beach Road along Lynn Canal. It will head up a hill and at the end of it is the Portage Cove trail head. It has had recent improvements done to it by SEATrails. The first half mile is wide and smooth. Past the point it's a trail with lots of tree roots. It will follow the canal and then from there you can continue on to Mount Riley.
Things to keep in mind: Though it is right out of Haines there are many black bears, brown bears, and moose on Mount Riley.