How I do Skills in 5e

Skills are a tool that drives the story forward, most DM's let you fail if you do not hit the target number required and I do the same but tend to give extra effort to well told stories, great plans, creative thinking and most importantly character classbackground as written.

Here is the way I use character class and backgrounds they wrote for there own characters,
For example, I assume you have Religion , History and Medicine as core skills for a cleric in D&D 5e. I take it that you spent much time reading your own and other Religions, local and far off history lessons and spent much time adhering to the sick and how to treat them.
So with that I assign 2 or 3 base skills to each class that was trained as an apprentice before graduating into said character class. This reaffirms a better qualification in said skills and drives the story forward when needed.

Mac Gruber in are campaign is a Human Cleric of  Sarenrae, He try's to do his obedience every day to show worship and to get that extra bonus to perception skill.😏
But enough of that back to skills, so far every time Mac has made a skill check based off of those 3 skills mentioned above I tend to give him advantage on the skill, especially in certain aspects.

The three skills I give advantage to this Cleric with the type of Cleric back ground, which is a generic cleric, this would change with say different clerics, ex. trickery,war,etc,domains.

Religion- If it deals with undead or good aligned gods or knowledge advantage is given.

History- as a local of Sandpoint all local checks are done with advantage of the town and surrounding area. All thassilonian checks as well due to the high volume of ancient artifacts in the area, and last to recall anything to do with Religion history is a choice between skills but at advantage.

Medicine- is not used all that often in my games but when it does come up, I treat clerics like battlefield medics that can diagnose how damage was done how best to treat a patient and most importantly anything to do with murder it can replace Investigation skill checks.

Another Example would be the Barbarian in are group,
For the Barbarian I looked at the skills they could choose from on page 47 of the players handbook and with the exception of perception that left 5 remaining choices. Animal handling,athletics,intimidation,nature and survival.
The Background story,

"Character Backstory I think that I am a half-orc of questionable / uncertain parentage. Abandoned shortly after my birth, I fended for myself and lived travelling from town to town looking for odd jobs as a day labourer, hence my illiteracy and dope physique (being ripped as shit). I’ve been taken advantage of several times throughout my childhood. What seemed like a generous and lonely benefactor, *named a major CEO in a company so deleted*, double crossed me and sold me into a forced labour camp, clearing trees and building roads (I have to decide on the area, but this plays into my Wanderer Feat and explains why I am so familiar with the surrounding geography / maps.) This experience also exacerbated my authority issues. That is why my character can be somewhat chaotic at times. I’m lashing out against authority, Marchionne, and the parents that abandoned me. Eventually, I escaped from the labour camp and joined the Bloody Skulls, a vicious tribe of full blooded Orcs. Despite my half-orc blood, I blended into the group well. Only one member, Dank Vaper, took umbrage with my parentage. We fought constantly. The final straw with Dank occurred one night when he attacked me while I was sleeping. He restrained me and then cut sweet, sweet, luscious hair off. I flew into a rage, escaped my bondage, and tried to kill him. The rest of the tribe separated us and eventually sided with Dank because of his full Orcish blood. The Bloody Skulls attacked me and left me for dead. To this day I look for signs of Dank Vaper to exact my revenge."

So with this I could easily not use animal handling, it was never mentioned, and never really intimidated anyone so in the end went with athletics,nature and survival for the character using the background provided to me by the character.
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Failure just makes other people attempt the same skill and it will go around the table if the next person fails, but have each character have a set of certain skills that they excel at gain advantage and push the story forward and drive home the fact that they are heroes and not dice monkey's rolling check after check.

I tend to not do this for perception unless it's important to the story to find something or I just fudge the rule and make it out in the open for anyone to find as it is so critical to the development of what needs to happen.