The nearest railway station is Palghat Junction (Palakkad). From Palakkad, go to Mannarkkad, and from there to Mukkali. Both places are well connected by private buses. If you are planning to stay at Mukkali, reserve rooms in advance at the Forest dept resthouse. Or else you can stay at Mannarkkad and hire a jeep to reach Mukkali by 8AM. Mukkali is about 60 km from Palakkad, with Mannarkkad en route. The visit to the Valley and Kunti river will take only a day, and you can return back to Palakkad the same evening.
Permission is required from the Kerala Forest Department. For a short one day trip (usually taken by tourists), permission can be obtained on the spot at the Forest Department office at Mukkali. This trip includes a short trek of about 2-3 km to the hanging bridge at the Kunti River.
The entry point of the road to the park is Mukkali. From Mukkali, it takes a 30 km jeep ride to get into the actual entry point of the park. Jeeps can be hired from Mukkali. It is difficult for cars to traverse, so don't attempt to drive unless you are adventurous and have an SUV and good skill on mountain roads.
Tourists are not allowed to stay in the valley. They must come back to Mukkali by evening.
Entry into the valley is allowed only from 8AM to 2PM, as you need about one and a half hours (each way) to go to the Valley and come back. The Forest Department deliberately does not allow going to the valley early in the morning to minimize disturbance to the animals. Tourists will not be allowed to stay/camp overnight at the valley.
You will not be allowed to proceed without a guide. A guide will be assigned to your group when you get permission from the Mukkali Forest Dept Office.
If you are just a casual tourist, then this is fine, but serious naturalists will have to contact the Wildlife Warden much earlier and get adequate permission. Trekking in deep forest (for a few days) requires permission from the Forest Department office in Trivandrum, and is difficult to get.
Anyhow, it is almost impossible to get permission to go anywhere near the core of the national park, unless you are a field researcher or something like that.