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?
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.
No, you cannot do so.
D&D 5e's rules do not grant a familiar as a class feature. All rules for familiars (aside from the warlock's pact of the chain class feature, which expands the list of available familiars) are contained within the spell find familiar. Find familiar allows you to call on a spirit that takes the form of an animal and accompanies you. It has some inherent magical properties.
On the other hand, your animal companion is a real animal who has bonded with your character. It is not a spirit and it operates on its own instincts (with some guidance from its ranger).
That being said, the concept of a familiar in 5e is not compatible with the animal companion class feature. Familiars are summoned spirits; animal companions are wild beasts. With some massaging of the idea, your DM might let you take some liberties with your animal companion but in general, this should not be allowed.