The PC needs modifiers to raise that 20 to 25 or higher
If you are trying to, for example, lockpick a very high-difficulty
lock that requires a DC 25 and you get natural 20, do you lockpick it
as it is "Natural Success" or do you have to have a +5 or higher
modifier with thieve's tools as well?
The core rules put Proficiency into play, as well as Expertise, by not having "critical success" for anyone trying a difficult task. For very, very hard tasks (above DC = 20) the "anyone can try it" approach will typically fail. For a PC with a skill proficiency, or double that (expertise doubles a skill proficiency) there is a chance for success.
- It is worth keeping in mind that anyone can try most ability checks even if not proficient. They'll generaly have a lesser chance to succeed than a proficient character.
Three modifier examples
My Ranger(Hal O'Jinya, lvl 5); Criminal background; proficient with Thieves Tools and Deception.
- Hal's dexterity is 18 (mod +4 to thieves tools). Level 5 Proficiency = +3. He can succeed on a Thieve's Tools based attempt with a DC of 27 if I roll a 20. If our cleric first casts guidance before the attempt, Hal succeeds on a DC 31 attempt if the cleric rolls a 4 and I roll a 20.
- Hal does not have proficiency in History. His Intelligence is 10 (+0). He can succeed in a DC 20 History check on a roll of 20, or up to a DC 24 History check if our cleric casts guidance and rolls a 4.
- Hal's Charisma is 8(-1), but he is proficient in Deception. He can succeed on a DC 22 Deception check with a roll of 20. If the Cleric's guidance roll hits 4, can succeed on a DC 26 Deception check with a roll of 20.
To make an ability check, roll a d20 and add the relevant ability
modifier. As with other d20 rolls, apply bonuses and penalties, and
compare the total to the DC. If the total equals or exceeds the DC,
the ability check is a success the creature overcomes the challenge at
hand. Otherwise, it’s a failure, which means the character or monster
makes no progress toward the objective or makes progress combined with
a setback determined by the DM. (Chapter 7, Using Ability Scores, PHB p. 174)
The only natural success on a natural 20 is on a to-hit roll: an attack roll. All natural 20 attack rolls are critical hits.
Rolling 1 or 20
If the d20 roll for an attack is a 20, the attack hits regardless of any modifiers or the target’s AC. This is called a critical hit, which is explained later in this chapter. If the d20 roll for an attack is a 1, the attack misses regardless of any modifiers or the target’s AC. (Chapter 9, Combat, Basic Rules, p. 76)
There is no critical success/failure when Using Ability Scores
There is some text in the DMG for "critical success" or "critical failure."
Rolling a 20 or a 1 on an ability check or a saving throw doesn't
normally have any special effect. However, you can choose to take such
an exceptional roll into account when adjudicating the outcome. It's
up to you to determine how this manifests in the game. (DMG, p. 242)
Note: saving throws also don't have critical successes, nor critical failures. But there is one exception: the unique rule for what happens on a 1 or 20 when rolling a death saving throw. While that is straying from your topic a bit, your comment under the accepted answer suggests that covering all three basic uses of the d20 would help you. The related question about rolling a natural 1 or 20 on a saving throws is here.