Rules as Written - Stay the MM
Familiars, Rules as Written, don't advance. The Pact of Chain Familiar doesn't improve stats over time. The only text we have on them is in Find Familiar and in the Pact of Chain class feature.
Pact of Chain
You learn the find familiar spell and can cast it as a
ritual. The spell doesn't count against your number of
When you cast the spell, you can choose one of the
normal forms for your familiar or one of the following
special forms: imp, pseudodragon, quasit, or sprite.
Additionally, when you take the Attack action, you can
forgo one of your own attacks
You gain the service of a familiar, a spirit that takes an animal form you choose: bat, cat, crab, frog (toad), hawk. lizard, octopus, owl, poisonous snake, fish (quipper), rat, raven, sea horse, spider, or weasel. Appearing in an unoccupied space within range, the familiar has the statistics of the chosen form, though it is a celestial, fey or fiend (your choice) instead of a beast.
Your familiar acts independently of you, but it always obeys your commands. In combat, it rolls its own initiative and acts on its own turn. A familiar can’t attack, but it can take other actions as normal.
When the familiar drops to 0 hit points, it disappears, leaving behind no physical form. It reappears after you cast this spell again.
While your familiar is within 100 feet of you, you can communicate with it telepathically. Additionally, as an action, you can see through your familiar’s eyes and hear what it hears until the start of your next turn, gaining the benefits of any special senses that the familiar has. During this time, you are deaf and blind with regard to your own senses.
As an action, you can temporarily dismiss your familiar. It disappears into a pocket dimension where it awaits you summons. Alternatively, you can dismiss it forever. As an action while it is temporarily dismissed, you can cause it to reappear in any unoccupied space within 30 feet of you.
You can’t have more than one familiar at a time. If you cast this spell while you already have a familiar, you instead cause it to adopt a new form. Choose one of the forms from the above list. Your familiar transforms into the chosen creature.
Finally, when you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell. Your familiar must be within 100 feet of you, and it must use its reaction to deliver the spell when you cast it. If the spell requires an attack roll, you use your attack modifier for the roll.
Upgrade with Magic
A sprite is powerful in stealth, has poison arrows, and great for flavor in a feylock. Unfortunately, it isn't very hearty. You can improve its armor (and weapons), or give it with temporary health points by magical means. The sprite familiar takes its existing equipment with it when sent to the pocket dimension, so in theory, the upgraded equipment should work go as well (there is a question on this, but even if the sprite couldn't take them with them, they are light and you can carry it). Magic items meant for PCs (isn't true of some of the weapons monsters carry) resize to the size of user, so there is no reason it couldn't use magic weapons/armor.
Ask your DM
Any more than this would require you get your DM agree to house rule for progression.