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: 
I will answer this on a generic level, as I have not gm'd a dnd game yet, so keep that in mind and fill in appropriate game system constructs for yourself.

In general, I dont feel like its a good idea to hand out experience points on a per enemy or interaction level. In other systems, particularly TDE, I will hand out the same experience points for everyone at either the end of an adventure or at critical points where it makes sense. 
Every player will get the exact same amount of experience as everybody else. Why? I have found that if I hand out experience individually, I risk to get several problems:

1. Kill-Stealing/Opportunity Stealing
2. Split of Power
3. Adventure Design

Now what do I mean by this?
The first point may be detrimental to cooperation. Your players may start trying to rush into situations without prior thinking, just to get extra experience, or similarly, trying to outplay each other in timing the moment to get more experience faster. Now this is probably not what will happen to you if your players like working together or if your encounter design is difficult. It may, depending on the "Social encounter" design, still occur though.

More pressing is a split of power. If a player happens to consistently have more opportunities than other players playing into the pcs strengths, a player may rush ahead in level of power, making balancing the encounters harder.

The most important factor I feel is mission desin and its impact on the "fun" of your players. I believe the GMs job is to make the games equally fun for everyone present. This also plays massively into the way you design your adventures. If you design them in a way that does not allow everyone to play important roles, the respective players will probably be dissatisfied afterwards. If it happens once, no problem, but more often...
So, If you want to do this, make sure you have have at least some opportunities for everyone to shine, at the very least you'd have more opportunities in the next adventure for those players disadvantaged in the current one.

In the end, its up to you, BUT I would advise on giving the whole group equal experience points and encourage the development of ideas and tactics more than actual mechanical performance. The only exception I would make is for good roleplay. Good ideas should result in an *overall* increase of experience to be handed out, not specific for individual players.

TL;DR:

If you have a competitive group or want to encourage it, while at the same time you're willing to spend extra effort on balancing each adventure in itself and towards the previous/next ones, go ahead and hand out individual experience points. 
As a modification for this, try not only to hand out XP for the actual interaction, but also for Ideas.

However, I'd pool up experience and hand out the same amount for everyone, except for special occasions where you feel good roleplay needs to be rewarded, similar to what @Jason K wrote.
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.