If Ranarion is my most personal 3e character, Arkteus would be the 4e equivalent. When we started our first 4e campaign, we were only 4, so everyone made a character, I took the spot as the main-DM, someone would do the substitute DM, so I can actually play my character at times (since I don’t believe in playing your own character in rules while DMing).
Arkteus is a wise-cracking tiefling wizard, who came from a nomadic tribe, left due personal reasons, joined a devil-cult afterwards, realized that he made a huge mistake and killed the cult’s head in his sleep, fled afterwards and tried to enlarge his horizon of magic in the most likely most advanced city in the arcane arts in this home-brew campaign. But they didn’t let him in (or any other strangers), so he loitered 2 years in-front of these gates, which would never open to him. But he wasn’t lazy, he studied the arts of his tribe, the magic the cult was using and grasped how arcane might is in the whole world. After this enlightenment, he came back to the civilization and started a career as an adventurer, to finance his further studies, which would mainly include the planes.
His attack magic was mostly fluff-oriented (after some months) and used temperature as a basic. So he had mostly fire and cold attack powers, I took even the feats to add these damage types to my paragon class (Bloodmage) powers and in the end got even some mind-controlling in there (thanks Tiefling supplement book).
After reaching 30th level and finishing the grand final of the whole campaign, he secluded himself once more. After years of absence, a party-member visited his tower, which was abandoned and only contained a single note: “I’ve created the doorway.”
In 5e you can’t combine all these fluff-shenanigans as clearly as in 4e. And one of the most important choices would be the Arcane Tradition (even though it won’t matter at 1st level, this is a very important for planning). I decided to go with Evocation, since the spells which got the most screentime were all these cold and fire spells. For ability scores, I used the standard in 4e (16, 14, 13, 12, 11, 10) and the assignment used the fact, that you could actually dump DEX as long as your INT was high. Since I used standard then, in this conversion I use the 5e standard and even if it might be not as smart as in 4e, I stick with the arrangement that time (so in the end 8, 10, 13, 15+1, 14, 12+2).
Backround is Hermit, since this might be the most important time in Arkteus’ turbulent life.
The spells and cantrips mostly have all the stuff Arkteus had or at least liked to have at that time.
For further levels: Elemental Adept (fire and/or cold) would be a fine choice, while Magic Initiate (warlock) would be like my start-feat in 4e which made Arkteus multiclass into the warlock class. Skilled for more knowledge skills would be appropriate, too, but in the end, Arkteus would definitely try to max out his Intelligence.
Take Evocation as your Arcane Tradition.
And along the line, my two signature spells in 4e were Arkteus’ Burning Blood (Bloodpulse with fire keyword) and Arkteus’ Mindburn (variation of the 20th level Bloodmage power, which got the fire keyword with a feat and instead of stunning dominated the enemies). Maybe I make my own versions of these spells in 5e at times.