My favorite short jungle hike is to Manoa Falls in back of Honolulu. The rain spills over from the windward side, which is why the mountains in back of Honolulu are so green. Drive on Manoa Road (not East Manoa Rd) to the very back to the trail head. You will pass bamboo forests, giant ferns, flowering ginger groves and the occasional orchid. This is the highest falls on Oahu, with a little pool at the bottom. About 2-3 hours round trip, including stopping at the waterfall. All tropical jungle trails will be muddy, slippery, sticky and humid but soooo beautiful.
There is a longer more "moderate" hike to a tropical fall and pool that is well-known called Maunawili Falls. It is in the back of Kailua, along the mountains. You have to cross over the stream four times, and you WILL fall in (at least your feet). This is an amazing fall and tropical pool in a lava crevasse surrounded by giant ferns, right out of a movie. You will see people climbing to the top of the falls and jumping into the pool here.
There will be a sign when you hike to many jungle falls and pools that say "no swimming"...but there will be people swimming in them; be careful because the reason it is disallowed is the presence of a parasite "leptospirosis". Also, if you do rain forest and jungle hiking, you absolutely must wear Tevas or very sturdy flip-flops. It is extremely muddy, and tennis shoes or hiking shoes will be ruined. Make sure you get good quality, because it will be no fun at all if your cheap flip-flops break in the middle of the jungle (you'll see a lot of broken ones on the trail...how do people hike in the jungle barefoot?). Also wear repellent.
Do you mean "windward shore" rather than "north shore"? 'Cause that is the side that is green and tropical, where the Koolau Mountains are sheer and plunge near the coast. The hikes above are my favorite, but if you absolutely want to see the wettest rain forest, then drive past Kaneohe going counter-clockwise back towards North Shore. Before Lae (Polynesian Cultural Center) you will swing around a fairly large bay (Kahana Bay). Kahana State Park is at the back of the bay and the start of the trail head. This is the rainiest part of Oahu, and this can be treacherous so be careful if you hike it. There are also homestead Hawaiians living next to the trail areas, so make sure to respect the "KAPU" signs ("taboo" signs in Hawaiian mean "KEEP OUT").
Like the other respondent, I think the trail with the most beautiful vistas is probably Makapu'u Lighthouse. From Honolulu, passing Hanauma Bay National Preserve (and hopefully you have snorkeled at this amazing place, great sea turtles, moray eels, etc.) you will pass Sandy Beach and things will look VERY dry. As you start to climb, there will be a parking lot to the right, which is near the start of this trail head to the old lighthouse. Amazing views, you can see Molokai easily, as well as look back down the coast to Waimanalo all the way to Kaneohe Bay. Humpbacks always seen in season (getting too late now, they are on their way back to Alaska and Canada). Can be hot, wear sunblock and a hat.