This claim has two parts: an explicit claim (that the CIA has a deal with Jeff Bezos), and an implicit one (that this creates a conflict of interest).
The explict claim is FALSE
The CIA has no deal with Jeff Bezos. They have a deal with Amazon, which is a publicly traded company.
Sources (swiped verbatim from ChrisW's excellent answer):
The current owner of The Washington Post is Jeff Bezos, who also founded Amazon.
News from Fortune.com (among others) in 2015 reports,
Intelligence community loves its new Amazon cloud
Two years ago, the CIA selected AWS to build and run a special, secure cloud to be used by 17 intelligence-related agencies, in a contract valued at $600 million. That was a watershed event for Amazon, the leader in public cloud services.
The implicit claim is unproven, and probably unprovable
It seems likely to us (us being the non-billionaire users of this site) that it certainly creates the appearance of a conflict of interest for Jeff Bezos, since he owns big chunks of both Amazon and the Washington Post. And we can assume that if the CEO of the paper chooses to get involved, the writers and editorial staff of the Post might feel a conflict.
But there's no proof of that actually happening. For an intervention in the Post's policies to occur without us knowing about it, dozens of people are likely to have to agree to suppress the truth.
Should the Post acknowledge this linkage?
Let's do a thought experiment: does Jimmy Dore acknowledge who contributes to his support? If some group were to have bought large blocks of tickets or CDs, or to have committed to underwrite a show, should he have to?