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This is actually a three-part question:

1. As a level 5 Tiefling Pact of the Chain Warlock, can I cast my Darkness spell on my Imp familiar?




2. If yes, once cast, would my Darkness spell be ended if my imp traveled 65 ft away from me?




3. If you can cast Darkness on the imp, could you maintain concentration on Darkness while using the Imp's senses?

1: No, but you can use an object to carry the darkness.

The darkness spell targets a point in space or a point on an object that you are carrying or isn't being worn or carried, so an imp isn't a valid target.

But you could cast it on a pebble, then give the pebble to your imp and let it carry that, and you'd get the same net effect. The darkness moves with the object you've cast it on, so your imp can carry the object around and be surrounded by a cloud of darkness.

There is some disagreement about whether a darkness- or daylight-enchanted object can be moved by somebody other than the caster without messing up the spell, so you might want to check with your DM, but the general consensus (and my personal interpretation) is that the clause about the spell "moving with" the item it's cast on makes it clear that the intent of the spell was to create an item that somebody else can then carry around with them.

2: No, range only matters during the initial casting.

The rule here is found in the last line of the Range section of the Player's Handbook spellcasting rules (p.202-203):

Once a spell is cast, its effects aren't limited by its range, unless the spell's description says otherwise.

3: Yes, concentration only ends under specific circumstances.

Again, in the Player's Handbook spellcasting rules (p.203), it tells you specifically what ends concentration:

• Casting another spell that requires concentration (even if you choose not to concentrate on the new spell)


• Taking damage (if you fail a save)


• Being incapacitated or killed


• DM-selected environmental effects (if you fail a save)

And of course any other effect that specifically says it can cause casters to lose concentration.

Sharing senses is an action, but it isn't a spell and doesn't say it requires concentration, so it doesn't cause any difficulties.

No.

Darkness: Magical darkness spreads from a point you choose within range to fill a 15-foot-radius sphere for the duration... If the point you choose is on an object you are holding or one that isn't being worn or carried, the darkness emanates from the object and moves with it.

An imp is a creature, not an object.

In short, no

Darkness must target...

a point you choose within range

... and a creature (your imp) is not a point.

What about something the imp holds?

note: this is only one ruling; your GM may rule that another creature can move the object, in which case you can give your imp the object you cast darkness on to carry (see the other answers on the linked question above)

There is mention of targeting a point on an object, so you might suppose you can target an object the imp is holding...

If the point you choose is on an object you are holding or one that isn't being worn or carried, the darkness emanates from the object and moves with it. 

... but you have to be holding an object (or no one can hold it).

Once someone else picks up the object, the point is no longer on an...

object ... that isn't being worn or carried [by someone other than you]

... and the darkness ceases to emanate from the object.

note: the darkness will still be there, it just won't be tied to the object anymore, so moving the object no longer moves the darkness

Moving out of range

If you were to target a point or object (or if a GM overruled so you could target a creature), then moving far enough to be out of the spell's range would not end the spell. From the Spellcasting rules:

Once a spell is cast, its effects aren't limited by its range, unless the spell's description says otherwise.

Concentration

There is no reason perceiving through your familiar's senses would explicitly restrict concentration. Only a few things can end Concentration (apart from willingly ending it):

• Casting another spell that requires Concentration...




• Taking damage...




• Being incapacitated or killed...

Perhaps a GM might...

decide that certain environmental phenomena, such as a wave crashing over you while you're on a storm-tossed ship, require you to succeed on a DC 10 Constitution saving throw to maintain concentration on a spell.

And use the blinding of your own senses as an environmental phenomena, but that is GM-dependent.