(do you like camping outdoors?) is used in America and it means do you like camping in the outdoors, there are some differences between British and American way of speaking, and that is what they say in America

Both are perfect... "Do you like camping in the outdoors." = This gives the impression of being out in the countryside or in the mountains! It's a great way to say it. If you were to say... "Do you like camping outdoors?" it would be asking: "Do you like camping inside or outside?", which is a stupid question 99% of the time. So you are 100% correct with your sentence. You might also say: "Do you like camping in the great outdoors?" = It gives it even more of a countryside / mountain feel :)!

The first one should be "Do you like camping outdoors?". The second sentence is correct.