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.