Tools (p.154 PHB) says:

A tool helps you to do something you couldn't otherwise do ... Proficiency with a tool allows you to add your proficiency bonus to any ability check you make using that tool. Tool use is not tied to a single ability, since proficiency with a tool represents broader knowledge of its use.

Disguise Kit (p.154 PHB) says:

Proficiency with this kit lets you add your proficiency bonus to any ability checks you make to create a visual disguise.

So, to make a disguise you need a disguise kit - no disguise kit, no disguise; you may be able to improvise one (with disadvantage) if your DM is kind.

You can use a disguise kit on yourself or somebody else, whoever you use it on uses your check to determine how good the disguise is.

You do not need to be proficient; proficiency with it allows you to add your proficiency bonus (duh) to your ability modifier. Which ability? INT (getting details right), WIS (observing what you are copying) and DEX (applying make-up) are all contenders; STR & CON and (surprisingly) CHA - not so much.

Importantly, you are making a visual disguise. It allows you look different

Deception skill (p. 178 PHB) says:

Your Charisma (Deception) check determines whether you can convincingly hide the truth, either verbally or through your actions.

The Deception check allows you to hide the truth; logically this can be the truth that its you wearing a disguise. It allows you to act different.

Depending on the situation, looking different may be enough or acting different may be enough or you may need to do both.

For example, you may want to get into the castle disguised as the captain of the guard. The DM decides that this is an Intelligence (Disguise Kit) check and rolls secretly, getting say an 8 + 2 INT + 2 Proficiency = 12.

The guards have a passive Wisdom (Perception) of say 12. The guards on the outer baily are inattentive and the captain comes through several times a day so they make the check with disadvantage (12-5=8) - you pass through with a nod. The guards on the inner baily are more attentive their 12 matches your 12 so they are suspicious - the DM Calls for a Charisma (Deception) opposed by their passive Wisdom (Insight), with advantage because the disguise is nearly good enough on its own. Now to get to the treasury, the guards are hyper-vigilant and advantage mean they see serious flaws in the disguise - make your Charisma (Deception) with no advantage or disadvantage against their passive Wisdom (Insight) with advantage.

Without proficiency in the kit, your total would be 10 and you would be questioned at the first gate rather than the second.

Now you could dispense with disguise altogether - claim that you are the captain polymorphed! Now, this would use Charisma (Deception) only but I can see a lot of disadvantage (literally) with this approach.

Alternative, you could disguise the charlatan sorcerer in your party who has proficiency with deception and high charisma; which you don't - you socially inept clod :). Here you are the make-up artist and she is the thespian.

Please note that the example is deliberately simplistic and there are whole nuances of difficulty I'm glossing over. There is a range for both disguise and deception ranging from looking like an Orc from half a mile away to convincing the queen of the orcs that you are her mate while you are, you know, mating.

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.