Question: So I'm creating my first D&D 5e campaign and I want there to be an even balance between combat and non-combat with a slight bias to the non-combat style of missions. However I'm unsure about how to deal out XP for these situations and how much is a suitable amount. Also should I some characters get more XP than others for certain missions? For instance if you're trying to steal an artifact from a temple then a rogue will be very good at this while a barbarian will likely not be able to do much and so deserves less XP than the rogue but is this a fair way of managing things?

Answer: 
The DMG gives you several alternate XP/leveling options on pages 260-261.

You may benefit from "milestone" leveling rather than doling out XP per encounter. Thus characters level up when they accomplish story specific events, so the exact path they take isn't dependent on being murder hobos. You can also get rid of XP altogether and just level characters up when you feel appropriate. 

Characters can earn XP for defeating creatures via negotiation, stealth, or intimidation, not just for direct combat (called "noncombat challenges" in the DMG) if YOU deem it so. You should explain this up front and possibly "over reward" non-combat interactions initially to get players in the habit of looking for non-combat solutions. If YOU only give out XP for killing things then expect your players to try to kill everything. Train your players.

It sounds like you may be planning to award XP to characters based on individual achievements (thus the barbarian you mention wouldn't get XP if the rogue does all the thieving). Collective XP will let the barbarian share in XP for things when rogue did the majority of work, just like the rogue can get XP when the barbarian slaughters a room full of guards. Many GMs add up the total XP awarded and divide by number of players, then possibly add some individual character XP awards for good roleplaying, MVP, etc, rather than try to attribute XP to each character for every specific task. The DMG also addresses allowing characters to level up at different rates, though this is mostly due to a player missing a session rather than not contributing as much in a particular session.
I play RoleMaster but the idea of XP is similar. 

The answer is in the word. I give experience points for relevant **experiences**. I dole them out (in order) by:

1. **Milestones**: Accomplishing a mission or reaching a certain plateau
2. **New experiences**: first time ever in town, picking a lock, meeting an owlbear, etc. (2nd time in a town, not such a big deal),
3. **roleplaying**: more on that later,
4. **Hits taken or received** (relative to monster difficulty),
5. **Spells** successfully thrown (relative to caster's level),
6. **criticals** (RoleMaster is a critical heavy playing system),
7. **ideas/specials** ("It was my idea to trap the gelatinous cube in the giant
squirt gun."),
8. **kills**, and
9. **treasures** (treasures are their own reward, in my opinion, and players usually don't get any XP for gold and magic items).

Rearrange or add to the list based on what is important in your campaign.

I also, with each new level, give the players a blank "Experience Sheet" form which has areas for hits taken, given, spells, ideas, thing encountered, etc. It's their job to keep track of things they've done individually on these sheets and I collect them when I'm going to award XP. (It's funny what some of them will put down to wrangle extra XP sometimes!) When I really like the way a character is playing their character (i.e. a character sniveling in front of a noble, polishing his boots, acting as a decoy) I make sure that the player wrote it on their sheet or I write it on a master sheet I keep for just those occasions.

Because I place the group activities highest (just watching an Owlbear rip apart your fighter is an experience), the individual experience is fairly well grouped with the more immersed individuals ending up getting a little bit more.

Hope this helped.