I am running a 5th ed D&D game and it rocks on toast! I hope someday to play in one.
Until then, here is something I would like to try: A half-orc Barbarian! (gasp!).
Part of the reason for this is that 5e is simplified enough that I can stomach the rage here a lot more than I could the rage/fatigue in, say 3rd edition/pathfinder. First of all, for regular rage there is no fatigue (although there is exhaustion on the frenzy path). Second, the rage (and exhaustion) are greatly simplified: extra damage + resistance during rage, disadvantage on ability checks during 1st stage of exhaustion, etc.
Also, the half-orc seems to have racial abilities that are "high level barbarian-lite" and also stack with the barbarian class abilities.
Anyhow, creating it is easy. Use the array, put 14 str, 15 con, 13 dex, 12 wis, 10 chr, 8 int. Racial mods put str and con both to 16 (this makes my unarmoured ac 14, which is just fine at 1st level). Take the outlander background. For skills I would get intimidate (free), athletics, perception, survival and stealth (if you double up a skill from class/race and background, just pick a new skill. Very handy!). The background also gives me proficiency in a musical instrument (flute), which I could purchase with 2 of the 10 gp the background gives me. Start with Great axe, shortbow (presumably with some arrows) and four javelins, an explorer's pack, a hunting trap, a rabbit's foot (my first kill!), travelling close, a belt pouch and a staff (in case something needs bludgeoning, I guess). No armour or shield needed, really (I want to fight with a great axe, which means no shield anyhow).
At higher levels, increase str by two twice, then con by two twice (maxing them out to 20), then at 19th level instead of a +2 stat improvement take the resilient feat to increase dex by one and get proficiency in dex saves (very nice to get a +6 boost at high levels, and also fits a theme of "good physical saves"!). This would bring the 20th level stats (with the barbarian 20th level bonus explicitly breaking the "20" cap on abilities that otherwise exists in 5e) to str 24, dex 14, con 24, int 8, wis 12, chr 10. So now my unarmoured AC is a respectable 19 (quite high in 5e). Take the berserker path but don't get stupid with overusing the frenzy ability (I would do it no more than once per day and save it for the "boss monster" usually - it is very hard to remove multiple levels of exhaustion). For most encounters, rage alone will do.
If I came in at 20th level, I note that 5e would give me wayyyyy fewer magic items than 3rd ed in a standard campaign. First of all, I would have about 21500 gp, but no guarantee of buying any but common magic items like potions of healing/climbing. Second, I would (for the "standard" campaign) get assigned one rare and two uncommon magical items (say a +2 great axe, a javelin of lightning (very cool in 5e), and a periapt of wound closure). I am fine with that, but those coming from a 3rd ed background are going to experience a bit of a shock.

Let's see, chaotic good, speaks common, orc and giant (extra language from outlander background, seems to fit - orcs and ogres interact sometimes). A child by marriage alliance between an orc and human tribe, he was taken by his mother far away when the alliance broke down and the tribes fell to war again (his father was likely killed in the battle). He lived in the wild with his mother Belloca as a hunter-gathering tribe of two, but after she died he was on his own. He resists the lure of Gruumsh (that urges all orcs and half-orcs to be evil) and sees the party as a new, better "tribe" that he can watch over. He has had visions of something nasty happening in the future, and tends to drink to keep these visions away. He believes that change is the only constant in life, and is in the habit of wandering (some might call it wanderlust - he just doesn't feel comfortable staying too long in one place).
Keth is ready to smash! The critical hit rules in 5e have you double the dice (in the case of a great axe that means 2d12 + X instead of 1d12 + X). But half-orcs add another weapon die for 3d12 + X, and at high levels, barbarians just add more dice. So a natural 20 (no need to confirm the crit in 5e) becomes
nasty.
Anyhow, this would be a "cut loose" character, and maybe I will play it sometime. It is a barbarian with less paperwork, which is fitting, in a meta-sense.