How would you build these characters in 5E?

[ZenBear]

http://i.kinja-img.com/gawker-media/image/upload/s--VJ9VpO95--/17n7yg9p8eemnjpg.jpg

With a combination of optimization, insofar as possible, and roleplay. Accept some changes; you don't need Tesla to have an actual lightning gun, he can cast lightning bolt instead. Alternatively, make it a magic item.

IMO: Tesla the Artificer, obvious. Lovecraft... Rogue with high INT and Expertise in Arcana, History, Religion, Investigation. Or Divination Wizard. I want to say Warlock GOO Pact obviously, but Livecraft was not Charismatic, by all accounts. That being said, homebrew swap CHA for INT and I'm onboard.

[Falcon X]

First off, all magic could be reflavored as technology if you wanted. Whenever a character learns a spell, instead it's just an invention they made like a lightning gun (lightning bolt), pheromone spray (charm), summon monster (teleportation device from unknown origins).

You might also consider storm sorcerer for Tesla, though a modded wizard is pretty good.

I'm on board for Lovecraft being an INT based warlock. Just get DM approval. Shouldn't be hard as CHA is generally better than INT.

[Oramac]

storm sorcerer for Tesla

This is what I would go with as well. The metamagic stuff could be modified ammunition for the Lightning Gun, or something like that.

[Thrudd]

You wouldn't. D&D isn't really made for that genre. You would have to strip the game of its implied setting and rewrite the classes, treasure, and most of the monsters to fit the pulp/steampunk/lovecraft mythos thing.

[Reaper34]

i'd switch systems to hollow earth expiditions by exile games and go from there.