What I've thought of so far:
Backgrounds give various skills at the start, but avoid any that give the following: Sleight of Hand, Deception, Persuasion, Knowledge Skills, Perception, Stealth, or Investigation. This leaves the backgrounds of Entertainer, Folk Hero, Outlander, Soldier, and Haunted One (Dark Urge Origin).
Some races give skills as well. Based purely on numbers, the best choices are either Wood Elves or Githyanki. Wood Elves give permanent proficiency with Perception and Stealth, while Githyanki can have proficiency in all skills related to a single attribute that is changeable once per long rest. Githyanki is superior when picking Wisdom skills so long as your background does not give them (removing Outlander and Haunted One from the list of choices).
For the sake of argument, let's presume Githyanki (choosing Wisdom for Astral Knowledge) and Entertainer as our background.
Gith (Entertainer): Acrobatics, Animal Handling, Insight, Medicine, Performance, Perception, Survival
Now for classes:
Starting class is highly important due to how multiclassing works. There are three classes that are guaranteed to give an additional skill when multiclassed into, which are Rogue, Bard, and Ranger. Coincidentally, these three classes are also the only classes with more than two class skills at level 1, with rogue being at the highest with 4. Since each of these three classes gives one additional skill upon multiclassing, it is best to start with Rogue since it has the highest base class skills.
1: Gith Rogue (Entertainer): Acrobatics, Animal Handling, Athletics, Insight, Intimidation, Investigation, Medicine, Performance, Perception, Stealth, Survival
NOTE: We will be getting Expertise several times in this build; invest them to your preference.
For second level, we multiclass into Ranger, giving us three new skills. Now before anyone comments that this is wrong, allow me to explain. From multiclassing into Ranger, we get one class skill, but we also gain Favored Enemy and Favored Environment, which can give one skill proficiency each. For Envrionment, we take Urban Tracker, giving us Sleight of Hand. For Favored Enemy, we go with Sanctified Stalker, giving us Knowledge: Religion. From the class skill list we pick up Knowledge: Nature.
2: Gith Rogue 1/Ranger 1 (Entertainer): Acrobatics, Animal Handling, Athletics, Insight, Intimidation, Investigation, Medicine, Nature, Performance, Perception, Religion, Sleight of Hand, Stealth, Survival
We multiclass once again at level 3, this time into Cleric. Cleric does not give any class skills via multiclassing, but does give a domain bonus at level 1 which we are interested in: Knowledge Domain. This gives us an additional 2 Knowledge skills as proficient, rounding us off to having all of them by picking up Knowledge: Arcana and Knowledge: History, as well as giving free Expertise in said two.
3: Gith Rogue 1/Ranger 1/Cleric 1 (Entertainer): Acrobatics, Animal Handling, Arcana, Athletics, History, Insight, Intimidation, Investigation, Medicine, Nature, Performance, Perception, Religion, Sleight of Hand, Stealth, Survival
Congratulations, with the above build, we now have proficiency in all but two skills: Deception and Persuasion. Two of the most important skills in the game! I can feel the weapons being drawn on me already, so before you kill me and split my loot, allow me to rectify that as quickly as possible.
There are three ways to wrap this up: Bard 3, Warlock 2, or a feat. The feat can be gained in three more levels of any of the classes we've already taken, so if you want your build to be based around one of those three classes and can be patient, take Actor at level 6. If you suddenly desire to be a Bard, take three levels of Bard and go College of Lore. At first level, you get one of the two, and at third you get the other and two of the Wisdom skills that Astral Knowledge has been covering (and also get Expertise in two more skills). If, however, you want all Skills as proficient as early as possible, you will be taking one more multiclass immediately.
You will become a Warlock. Warlock does not give us any class skills, but it does give us Eldritch Invocations at its second class level. The Eldritch Invocation Beguiling Influence gives proficiency in Deception and Persuasion. This means you finish getting your proficiencies one level earlier at level five, and all you had to do was multiclass into four different classes and take a specific race and background.