There are three main causes of chain induced vibration. More correctly, the chain is not the initial causal factor, but a resultant one passing the grief down the line as it reacts to other improper conditions such as chain misalignment, wear, and incorrect adjustment.
Chain misalignment causes grief to the unforgiving sprockets as the chain grates against the inner or outer sprocket, as the case may be, cutting a new path on the teeth. This causes vibration and a plethora of metal fragments that wreak havoc on the moving parts. Misalignment of the sprockets will cut a new path on the inside or the outside of the sprockets depending on which sprocket is misaligned. Of course, this will also destroys your chain. Wear occurs naturally with time and usage. Chains wear unevenly and when worn, will have tight and loose spots. A badly worn chain will cause a whipping or jerking action. Insufficient lubrication or extreme wear will cause links to kink and not fully extend.
Finally, the rear wheel misalignment which causes – you guessed it – premature wear, and sooner-than-later vibrational destruction may compound this.
There is a fourth possible, but improbably cause, which is one of the axle bearing failing. This does occur, but should be last on the list to check. The chain's resultant vibration will be a symptom of the sprocket shaft scissoring out or true. You can easily check the sprocket shaft bearing by trying to move the front sprocket... if you feel movement you have problems with the front sprocket shaft bearing.