The original plan was to make a Light Cleric, but I came up with a cool backstory that would better fit a Fiend Warlock. The problem is, that our party has NOT ONE character who has access to healing spells. The other players are: A Way of Shadows Monk, a conjuration Wizard and a cavalier Fighter. We are at level 3 and I rolled an 18 on my primary stat, so I could take a the magic initiate feat at level 4, instead of the ASI, to get healing word or cure wounds. Would that be a viable solution? 

TL,DR. I came up with a cool backstory, that would better fit a different class, but I am afraid of the total lack of healing spells. 

I'd say play what you want, unless the "original plan" was the result of the whole group creating characters together and you committed to a cleric - in that case you should probably still play what you want, but you should let the others know beforehand in case that changes things for them.

Quote from G3raltofR1via >>

The original plan was to make a Light Cleric, but I came up with a cool backstory that would better fit a Fiend Warlock. The problem is, that our party has NOT ONE character who has access to healing spells. The other players are: A Way of Shadows Monk, a conjuration Wizard and a cavalier Fighter. We are at level 3 and I rolled an 18 on my primary stat, so I could take a the magic initiate feat at level 4, instead of the ASI, to get healing word or cure wounds. Would that be a viable solution? 

TL,DR. I came up with a cool backstory, that would better fit a different class, but I am afraid of the total lack of healing spells. 

Your backstory should work for either - Hell has deities residing in it, like Lolth, and we're pretty certain some deities pact, like The Raven Queen.  If I were your DM, I would 100% support you being a cleric whose backstory was that you tried to pact with a deity and you came out the other side a cleric instead of a warlock, mechanically, but with identical storytelling surrounding it.  If you think a cleric would be more fun, just ask your DM if this would be ok.  I'm guessing they'll say yes.

Likewise, in terms of picking warlock patrons.  There is no reason the "Fiend" pact subclass has to actually be with an actual, literal Fiend in every setting - you can and should customize that to each story you tell.  Maybe you're mechanically the Fiend but your patron is Apollo, or maybe you're mechanically the Archfey but your patron is Delilah the Arch-Succubus.  If you want the Fiend subclass benefits, great, but if you wanted some healing instead, consider discussing with your DM having the fiend you pacted with in your backstory mechanically granting the Celestial benefits.  Guessing they'll say yes to this, too.

Either way: play what you want.  DnD with no healing support just makes it more challenging.  Don't take Magic Initiate to fix it, though.  Take Inspiring Leader and discuss tool proficiencies with the conjurer (conjurers have obvious synergy with pretty much every toolkit) - if the party has cook's utensils and herbalism kit proficiencies, you can lean into them if you find yourselves hard-up for healing.  You can get tool proficiencies from your background.

I believe you should always play what you like.  The goal is for you and your group to have fun and this works best if everyone has a character they enjoy.  
If you love your warlock backstory idea, then play a warlock.  As Lyxen said, 5e is designed so that a party is no longer dependent on having a full time healer.   Of course, if you are looking for a way to combine these ideas, you can also talk to your DM about multi-classing, or you could consider a Celestial Patron Warlock.  Their Healing Light ability is pretty awesome -- 60' range and works as a bonus action.

It's hard to say more without knowing the details of your back story.  If you want to share, I would love to read it.

Healing potions 

And play what you want. Discuss it with the group if it was already decided you were playing the cleric. But really you don’t need a “healer” to work.