It's worth to rent a motorbike for having a look around. It's not only that you can reach then a few places nearer by, but you can make great tours through stunning landscapes along the Mekong River. Both directions, river upwards and river downwards are worth to go.
Ban Fai at the mainroad rents motorbikes for 200 Baht to 250 Baht.
From Chiang Khong riverupwards it's the road to Chiang Saen, which splits up after a few kilometers. The straight one leads through the mountains, coming along at a Hmong village on a mountain top (some 15 km). Turning right there into the village and going up further leads to a Thai Lue village and offers stunning views over the surroundings and the Mekong.
The other road (right turn) follows mostly the river banks all the time and is one of the nicest trips you can do along the Mekong. When coming back from Chiang Saen, you can drive the other road and make the tour a loop.
It's much wilder here than in Thailands northeast, the Issan, because of the mountainous surroundings. The roads are all small and there is little traffic on them.
If you like going around by motorbike consider making a day trip from Chiang Khong to the Golden Triangle. The advantage of an own vehicle is that you can stop wherever you want. That means that you come really apart from the tourist tracks.
Another great trip to do is to follow the road 1155 via Lai Ngao (Wiang Kaeng) to Thoeng. The first part of the trip leads parallel to the Mekong River, which is the border between Thailand and Laos. Then the road leads through a long valley, getting smaller and screwing upwards into the mountains. To the left side are high mountains who are the border between Thailand and Laos here, to the left are also mountains. A shortcut is to turn right at the sign to Khun Tan. That means crossing the mountains and driving up a long and extremely steep slope. On the top there is a Hmong village. On the other side one comes back to the plains and turns right on the road 1020 back to Chiang Khong. Otherwise you continue to Thoeng and make the turn there back to Chiang Khong, also on the 1020. Alternatively you can leave the 1155 and turn to the left. That leads you up onto the mountainous border to Laos. A few stunning views offer Pha Phueng and Phu Chi Fa viewpoint. Few people live here and the mountaineos surroundings look wild and abandoned. You need a map for these trips. Maps are available in Chiang Khong's 7-Eleven Shops or sometimes at the small private Tourist informations who sell also tours and boat trips and so on.
Other tours around Chiang Khong are to go to Huay Meng and have a smaller hike to the waterfalls there.
Another waterfall is behind a Hmong village which is 10 km away from Chiang Khong in the mountains. From the village it's a tough 3 hours tour following a small river upwards, first crossing rice paddies, then entering the jungle and following a jungle path for a longer time. There is a natural pool for swimming. But, that means really jungle trekking and it's good to have good shoes and a machete with you.