This sounds like two things.
• Start at level 0, using background features.
• Advance to level 1 in a 'Classless' class.
The Classless class has something like a pointbuy system to purchase any feature from any class.
I did this Classless Class for 4e. It works well when features are gated by level prereqs. In 5e, certain features are very expensive, such as Rogue Sneak Attack, which is worth something like 1½ feats, thus require two levels to acquire, meaning, there will be an 'empty level' as downpayment to purchase it in the next level. 5e level 1 is front loaded worth more than one level of design space, in order to start with features like Sneak Attack.
I was thinking of a class that has features like this
Level 1: Destiny Points
Your belief in yourself places a special mark on the world. This is represented by Destiny Points. Your GEC level determines the number of points you have, as shown in the Destiny Points column of the GEC Features table.
You can expend these points to enhance or fuel certain GEC features.
You start knowing 2 such features: Defy Death and Lucky Hit, and each of which is detailed below.
Defy Death: When you are reduced to 0 Hit Points but not killed outright, you can spend a Destiny Point and drop to 1d12 Hit Points instead. You cannot exceed your maximum Hit Point total this way.
Lucky Hit: When you hit a creature with an attack and deal damage, you can spend a Destiny Point and gain a 1d12 bonus to the attack’s damage roll,
When you expend a Destiny Point, it is unavailable until you finish a Short or Long Rest, at the end of which you regain all your expended points.
Level 2: Learn by sight.
Your lived experience on adventure has granted you new aptitudes. You gain one Origin feat of your choice.
Level 5: Fated Strike
You seem to strike at the weak points of enemies more often when they let your attacks land. When you hit with an attack, you can spend a Destiny Point to make it a critical hit.