Q:

Can a ranger make their animal companion into a familiar? 

If I capture a creature/wild beast and train to be my companion - as the ranger class - is it possible within the rules of 5th edition d&d to give my new companion abilities that a familiar has, such as shared senses or telepathically connected?  Perhaps via paying a wizard to perform a ritual or other form of act? 

A:

Unless your DM explicitly creates houserules that allow you to make your companion into a familiar, no, there is no way to do this. 
The Find Familiar spell lets a wizard

gain the service of a familiar, a spirit that takes an animal form you choose: bat, cat, crab, frog (toad), hawk, lizard, octopus, owl, poisonous snake, fish (quipper), rat, raven, sea horse, spider, or weasel. 

Even more importantly,

it is a celestial, fey, or fiend (your choice) instead of a beast. 

The Ranger's companion must be a beast, and a familiar is not a beast. 
The Warlock's Pact of the Chain feature adds some potential forms to the list available, but does not alter the clause requiring familiars to be celestial, fey, or fiend. 
Outside of the Find Familiar spell, there are also a limited selection of creatures in the Monster Manual that have a variant sidebar that describes how they can be bonded to a spellcaster as a familiar.  However, none of these creatures are beasts, so none of them are eligible to become a Ranger's companion. 
Important caveat: The 1st sentence of this answer is partially untrue. 
While you can't make your companion a familiar, you can get some of the functionality of a familiar using your companion. 
The 2nd-level Ranger spell Beast Sense will allow you to see out of your companion's eyes and hear out of it's ears, just as a wizard can with their familiar. 
The 1st-level Ranger spell Beast Bond (from the Elemental Evil Player's Companion), will allow you to maintain a telepathic connection with your companion, just as a wizard can with their familiar, while also granting it advantage on attack rolls against enemies adjacent to you. 
Note that both of these spells require concentration, so you can't use both at once to fully replicate the functionality available to a wizard and their familiar.