The only class (or subclass) that gets the spell (with three exceptions) is the Wizard
This means that you will have to either dive into another class or take a feat, so let's list the options:
Taking the Ritual Caster feat
This allows you to cast rituals, though Clerics can already do this so you do not gain as much as some other classes. You would gain access to Wizard rituals that are not also Cleric rituals (alarm, comprehend languages, find familiar, identify, illusory script, Tenser's floating disk, unseen servant, magic mouth, skywrite, Leomund's tiny hut, phantom steed, water breathing, contact other plane, Rary's telepathic bond, and Drawmij's instant summons).
Taking the Magic Initiate feat
This is probably the worst option; the only real benefit it has over Ritual Caster is that it gets you two additional cantrips. Furthermore, with this feat you can only cast find familiar once per long rest. I would imagine that using a familiar mid-combat to deliver touch range spells increases their chance of dying and thus increases the need to be able to recast the spell with more regulatity than what this feat gives you.
Taking one level in Wizard
This is a strong option; it would get you find familiar, more cantrips, and the Wizard's Arcane Recovery feature; though it does delay your spellcasting by one level and being an entire level behind on getting higher level spell slots is quite the cost in and of itself. (How that cost compares to losing out on an Ability Score Increase will vary by table and likely depend on the party composition, style of play, and other factors.) This option also gets you access to the 1st level Wizard ritual spells (alarm, comprehend languages, find familiar, identify, illusory script, Tenser's floating disk, and unseen servant) if you happen to find them written down somewhere as you could copy them into your Spellbook.
Taking three levels in Warlock
By taking three levels in Warlock you can get a Pact Boon and, in particular, the Pact of the Chain:
You learn the find familiar spell and can cast it as a ritual. [...]
On top of Pact of the Chain already enhancing find familiar, this also grants access to the various Eldritch Invocations that enhance the spell even more in numerous ways.
Alternatively, you can take the Pact of the Tome and the Book of Ancient Secrets Eldritch Invocation:
You can now inscribe magical rituals in your Book of Shadows. Choose two 1st-level spells that have the ritual tag from any class’s spell list (the two needn’t be from the same list). [...]
This has the benefit of giving you a resource for ritual spells from any class, but the maximum level for those spells is based off your Warlock level and this requires the class combination of Cleric/Warlock which may be quite unlikely depending on your GM. This is also fundamentally different from multiclassing into typical casters as the spellcasting rules are entirely separate.
Taking three levels in Sorcerer
This doesn't actually get you find familiar but it does allow you to cast touch-range spells from a distance (by using a limited resource). By taking three levels in Sorcerer you can get the Distant Spell Metamagic:
[...] When you cast a spell that has a range of touch, you can spend 1 sorcery point to make the range of the spell 30 feet.
This would get you an additional second Metamagic option as well; though both would be limited by your number of Sorcery Points which are based only on your Sorcerer levels.
Taking the Trickery Domain Cleric
This is similar to the Sorcerer solution where you don't get the actual spell but can cast touch-range spells from a distance. Thomas Markov's answer explains this option in more detail
Taking three levels of Fighter or Rogue
Technically there is another way to get the spell which is to take three levels of Fighter as an Eldritch Knight (or three levels of Rogue as an Arcane Trickster); however, they cannot cast spells as rituals so you would be heavily limited by your total spell slots. That said, perhaps it would be better than Magic Initiate as it would get you Second Wind and Action Surge where the former allows you to heal yourself without using up spell slots and the latter allows for things such as casting flame strike twice in one turn.