As of this posting, the Cincinnati Bengals have seven picks for the 2014 NFL draft. There are some projections that are giving Cincinnati three compensatory picks for the loss of Manny Lawson, Pat Sims and Brad Gradkowski. It's possible that the Bengals receive a fifth-rounder for Lawson, but most likely it'll be a six and two seventh rounders. Regardless, awarding compensatory picks won't happen until later this spring and outside the scope of this discussion.
When considering the team's salary cap, you have to consider the rookie pool -- the amount of money set aside for rookies from the NFL draft (and a little more usually for undrafted free agents). Looking at salary cap projections, Cincinnati has an estimated $23 million of space against the 2014 salary cap. The general theory is that you'll need to remove roughly $5 million from those projections to account for 2014 rookies.
OverTheCap.com released initial rookie pool estimates this week. According to their numbers, the first-year pool is $4.5 million with an overall value of $24.2 million. Last year the Bengals rookie pool was $6.3 million for the year one pool, $9 million alloted for signing bonuses, with a total cap estimate of $33.75 million.