Easiest way?
I think I have something to offer here...
I, too, have been having this issue. When I click on an album, I only get the first song to play and playback stops. This didn't happen until fairly recently.
With regard to "checking" songs, I think there's some confusion. The second column from the left, with the little boxes, requires that all songs are checked in order for them to play back. but that's different than "checking" the songs for creating a playlist, for example, or an "up next"--I discovered this problem only after using my new Apple TV last night for the first time.
I had "unchecked" everything, by accident, really, a month or so ago....I used to have everything checked, and it would synch all checked songs to my iPod classic. Then, when I filled that up, and started putting songs on my iPhone, I left those new songs "unchecked" and loaded them, by album or artist, when synching my iPhone. I haven't synched my iPod classic in a while, as it's full and I just charge it every few days. So "unchecking" all songs didn't really affect me, until my attempt to playback an album at a time on Apple TV.
In order to do that, I had to add each song, one by one, to a "next up" list, or create a playlist, by choosing each individual song in order, one by one. It was maddening, and I feel for those who are frustrated by this.
I take it that by checking all songs, via the second column in iTunes, this problem will be resolved?