My cleric was convicted of a crime, before the became devout, and sentenced to a life in the Monastery of the Death God. She started with the grim, reclaiming bodies from war, plague or nature, burial duties and general clean up duties around the morgue. They soon realized that not only did she have the stomach for such repulsive work, but she was beginning to embrace it. She was promoted to work with those living near death, hospice shifts and taking care of the dying outside of the monastery. After years of increasingly reliable actions, learning rites and rituals, becoming more devout, her belief in her god solidified. All these thing led up to her training to become a monastic representative, a cleric of the god of death.
A lot of races work well with this idea in mind, numerically only a few give points to wisdom, Aarakocra, Protector Aasimar, Hill Dwarf, Wood Elf, Firbolg(+2), Water Genasi, and Half Elf or Human. Unofficially there are Zendikar Elves (+2, Planeshift: Zendikar) Pacifier Bearfolk, Centaur, Gearforged, Savage Gnolls and Nightwhisper Trollkin (All from Midgard Heroe's Handbook)
I tend to like Aasimar and Half Elves, but again, +1 wisdom is not all that much depending on your stats and how you get them. I'm going to appeal to as many people as I can and stick with PHB+1, so we'll go with something out of my comfort zone, Hill Dwarf. I'll choose mason tools, maybe she was a white collar criminal forging carvings.
Using point buy, Strength 12 for a bit of melee, 14 Dexterity max bonus with medium armor, 14 Constitution (+2) for HP and concentration checks, Intelligence 8 dump, Wisdom 15 (+1) main stat, and Charisma 8 dump. Again these are going to be different if you don't want to min/max; but why not be the best?
We'll be using the Grave Domain from Xanathar's Guide to Everything. We get 2 proficiencies, I'll pick Insight and Medicine both wisdom.
Acolyte background gets us Religion, and I'll choose Survival to help me live in between adventures. Two languages useful to your campaign. I'll take Giant and Celestial from my studies. Traits: I’ve spent so long in the temple that I have little practical experience dealing with people in the outside world. and I see omens in every event and action. The gods try to speak to us, we just need to listen. Ideals: Tradition. The ancient traditions of worship and sacrifice must be preserved and upheld. (Lawful) I can relate this to my Giant language and studies. Bonds: I owe my life to the priest who took me in when my parents died. I'll change priest to temple and dead parents into incarceration. I owe my life to the temple that took me in when I was incarcerated. Flaws: I put too much trust in those who wield power within my temple’s hierarchy.
Alignment: Lawful Neutral, lifestyle: Poor.
Warhammer is better than mace, scale for armor, a light crossbow is better to use because of our dexterity, a priest pack, and a shield and an amulet for my holy symbol. Acolyte gives us some more choices, I'll take a reliquary and prayer wheel to keep in it. I like to have some standard adventuring gear, ball bearings, pitons, rope, 10 ft pole and grappling hook totaling 4g 1s.
Since we're in XGtE, we'll take Toll of the Dead, but chill touch is another great thematic choice for damage. Thaumaturgy is a must as is Mending (now we can cut up that 10' pole into manageable parts!) Guidance and light are also some go to cantrips. Cure wounds, Healing Word, and Bless are staples of this class, in addition I will grab Protection from Good/Evil for the undead theme.
And there we have it, my take on the Grave Cleric.