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.
It as been awhile since I have been able to DM, however, I have always believed in a very simple system. Most of what I was taught growing up with the game is posted here, just in different variants. 

I have always taken the campaign as a whole. The completion of the campaign creates x amount of experience. 

Roleplaying is required by all members. Those who actually role-play their characters get experience points for doing so. Those who are along for the ride will find the struggle to level up harder.

Character Class tasks: A fighter is expected to fight, wizard expected to fight with magic (offense and defense) or use their spells to detect the out of the ordinary things In the world. Clerics are expected to fight, heal and protect from the supernatural. Rogues are expected to fight when it is advantageous to them (sneak attacks, back stabbing, etc...), detect traps, remove traps, pick locks, etc...

These tasks are rewarded on a simple method. If they are a bystander and watch, they get experience points for the experience (small amount) but those who actually are involved in the character class task get rewarded for success or failure with more xp for success. 

Attacking and killing creatures is part of the total campaign xp. I truly believe in the keep it simple stupid method. (KISS).