I'm working on a Corporate Directory at my institution and am trying to figure out some customization pieces. I know of how to customize the People Item template to add additional fields and had no issue with this.

I'm trying to replicate a display that's similar to a Staff Directory offered by this company:

http://www.vinewave.com/Products/StaffDirectory.aspx

So rather than displaying each result on a single table row, I would like to display 3 items or so in a single row, to replicate a grid like view of the People Item div containers. 

I assume that this probably has to be done by creating a custom control display template to handle this rendering. If this isn't possible, then I assume the next step would be developing a custom App to be deployed. If someone happens to have some resources that can point me in the right direction or examples of how to, I would greatly appreciate it. I feel like I've nearly searched the end of the Internet looking for answers!

You would be able to recreate this setup using a customized control and item display template as part of a content search web part (or search results).  I don't have any direct code samples to replicate the grid but I have used this template to start from and it would be give you the last name filtering and 2 new templates that you can begin customizing to be in a grid.

http://sympmarc.com/series/create-a-simple-sharepoint-2013-employee-directory-on-office365/

Here is a jsfiddle that shows how to separate a list into columns.

http://jsfiddle.net/kevinPHPkevin/MKL4g/131/