No, a Kenku will never be able to speak fluently

The Kenku were cursed by their previous Master after betraying it. The curse came in three parts (Volos PP 109):

1. The wings of the Kenku were withered and fell away, leaving them bound to earth.
2. The spark of creativity was torn from their souls.
3. Their master took away their voices.

In addition, they can only be Hopeless Plagiarists (Volos PP 110)

Although unable to speak in their own voices, kenku can perfectly mimic any sound they hear...kenku cannot create new sounds and can communicate only by using sounds they have heard. Most kenku use a combination of overheard phrases and sound effects to convey their ideas and thoughts.

The phrase highlighted above further shows that while a Kenku can read and write Common/Auran, they can only communicate with others via Mimicry, and that mimicry is limited (no passing notes in class!)

On page 111 of Volos, there are further directions for roleplaying a Kenku that describe how you this can be done at the table.

The curse given to them by their Master has negated their ability for fluent speech. Their ability to mimic allows them to communicate, but not fluently. Fluent speech would be a removal of the third part of the curse, which has not, and likely will not occur by simply using their current capabilities.

The pursuit of fluency could join their pursuit of flight as a character/plot hook.

Additional insight from Jeremy Crawford

Jeremy Crawford gave some more insight into Kenku:

friend will be playing a kenku in a one-shot. We discussed how mimicry works. Are they more like a parrot, repeating what they've heard? or can they observe a group for a time and make sentences using said voice? how long can they remember sounds?

A kenku is capable of speech. But that speech is a bizarre mix of syllables, words, and phrases in voices the kenku has heard.

More about kenku: they can cleverly piece together voices and sounds they've heard to communicate. They're not less intelligent than other humanoids, but they do have an unusual form of communication. To simplify roleplaying one, think of two-three voices for the character.

These two insights continue to suggest that the Kenku is speaking by combining an amalgam of previous sound captures. This does not equal fluency, but it is more that the more they hear, the more complex sentences they can construct.

Increased efficiency and ability does not equate with speaking fluently, they are only able to mimic a wider range of information.

Anyone can attempt to do that

Anyone can attempt to do anything, whether they have the feat to do it or not (even try to recall everything accurately, as in Keen Mind). It doesn't mean they will always succeed, but it also doesn't mean they need the feat to make an attempt.

This is an example of bad wording, for a feat

The way this feat is framed implies that only characters with this feat can succeed. Drawing on the wisdom from the Unearthed Arcana: Feats, specifically on the "bad feat" (as identified in the article) named "Warhammer Master," we learn the following:

The ability to knock aside an opponent's shield is nifty -- but that's something any character should be able to attempt. Locking that down into a feat threatens to limit the game's flexibility. You could argue that anyone could still try that trick, but the way the feat frames the ability makes it sound like only characters with this feat can succeed. This option is an area that I'd want DMs to adjudicate on their own, rather than bloating the game with fiddly rules.

This is in response to this part of the Warhammer Master feat:

In addition, you can use your warhammer to knock away a foe's shield. If you hit a creature that’s using a shield, you can have the attack deal no damage and force the creature to drop its shield.

The design goal, as far as Mike Mearls was concerned, was to write feats in such a way that they did not limit the game's flexibility. The way this part of the Actor feat is written implies you need to have the Actor feat to even attempt to mimic someone's speech.

How do non Actors do it without stepping on the feat?

I believe the non-obvious but valid intent, which will still grant this ability to everyone else without necessarily needing to step on this feat's toes, was to imply that the Actor is so good, they can mimic anyone after one minute's worth of study. Everyone else has to take at least a few hours before they can even try properly.

Listen to a chirping bird for a minute, or a bubbling brook, and try to mimic that sound. You might be able to, but you probably can't; but, someone in the world probably could (out of the 7B people on Earth, chances are pretty high, right?)

This feat makes your character so good, they will find that person is themselves, for any sound or speech pattern.

What do they roll?

The Actor feat gives us the guidance for what skill to use:

A successful Wisdom (Insight) check contested by your Charisma (Deception) check allows a listener to determine that the effect is faked.

This makes sense as you are trying to deceive someone that you are what you are not. Based on this feat, then in most cases, you should roll Deception vs Insight. However, for non-Actors, you could use different skills as you see fit.

• A Charisma (Performance) check could replace Deception, if you interpret this particular check as a test of how masterfully they can mimic another sound. Perhaps they are openly mimicking robins singing to an audience of children.

• A Charisma (Stealth) check could replace Deception, if they are trying to hide by creating noise that makes them blend into the environment.

If interpreted this way, then for the Actors, the predictability of what skill to use means they know where to put their proficiency and Expertise, should they acquire them. This makes this feat even more useful, because for Actors, they always know which skill to use.