Is it viable to be able to multiclass being both a Warlock and a Cleric to the same being if that being is a deity?

I get the impression Patrons don't have to be deities but can be, and I wasn't sure if there was precedent for double tapping them as a power source so to speak.

If your DM is ok with it, then it’s possible. But usually, it doesn’t happen.
Typically warlock patrons are not gods. Gods have clerics who serve them willingly and enthusiastically, they don’t need to go cutting side deals with shady people looking for power.

Strictly speaking, you can be a cleric without serving a god, the divine magic comes from the domain. 

Typically this is facilitated by a god's intervention. But your relationship to a god is entirely up to your imagination.

You could make an athiest cleric if you really wanted to. 

So, can your warlock and your cleric classes get their magic from the same creature? Yeah if you say they do, they do. That stuff is all narrative fluff. 

A wise Kobold oft says "flavor is free."

As long as the DM okays it, your patron can be whatever you want.

Alrighty, thank you guys.

I assumed it was a matter of flavor but wasn't sure if there would be any mechanical roadblocks and wanted to be sure before bringing it up to the DM.

Whether you can have the same being as a deity and patron is entirely up to your DM. I'd ask them.

In addition, whether your cleric can be devoted to something other than a deity is also entirely up to your DM. Unlike most other classes, the cleric's basic nature is tied up in the building of the game world. Some DMs would allow a cleric to draw divine power from something other than a deity. The DMG contains examples of clerics obtaining their abilities from forces or philosophies pg 13. However, that is a world building rule and not a player rule and their is no requirement for a DM to change the PHB rules to accommodate it.

Personally, I wouldn't have any issue with a multiclassed character having the same being as both deity and patron. It might be interesting from a role play perspective watching a character trying to reconcile the two different views of their deity. It also might make sense for some deities that have multiple aspects. For example, a god of deception, trickery and mystery might revel in having both clerics and warlocks as followers.

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Finally, if you are looking for RAW from the PHB, a cleric has to have a deity and warlock patrons are not gods.

"Divine magic, as the name suggests, is the power of the gods, flowing from them into the world. Clerics are conduits for that power, manifesting it as miraculous effects. The gods don’t grant this power to everyone who seeks it, but only to those chosen to fulfill a high calling.

Harnessing divine magic doesn’t rely on study or training. A cleric might learn formulaic prayers and ancient rites, but the ability to cast cleric spells relies on devotion and an intuitive sense of a deity’s wishes."

"As you create a cleric, the most important question to consider is which deity to serve and what principles you want your character to embody."

A cleric's basic abilities are not dependent on the domain, they are dependent on the deity, the domain just specifies a few spells that they cleric gets as extras for being a particular type of cleric for their deity.

In terms of warlock patrons "The beings that serve as patrons for warlocks are mighty inhabitants of other planes of existence—not gods, but almost godlike in their power." They are called out specifically as not being gods.

However, a DM is free to change both of those in their game world.