How to determine ocean bottom depth for a data set consisting of longitude and latitude?
The easiest products that provides ocean bathymetry (bottom depth) is the one you mentioned: https://maps.ngdc.noaa.gov/viewers/bathymetry/. It provides bathymetry data from a plethora of surveys. If you click in the map at any location, you will get a decent estimate of the depth at that point. You can also extract the surveys and data that are available for any given location. You also get DEMs and Lidar data. You can define a bounding box or a polygon (Identify tab) and it will provide the links to the original data. The original data in areas where no recent surveys are available is usually either from ETOPO or GEBCO. 

If you are interested in a specific area you can also use a region selector directly from  GEBCO (pick a box around all your points and extract the data that way).

You can also try NOAA electronic navigational charts.