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.
You can also award XP based on how important the non combat event was toward progressing the general adventure. A technique I have found to work fairly well is to take the party's average level as a base CR. Tiny inconsequential events (ex: talking to a shopkeeper in character) are worth CR minus 2 as an experience award. Moderate events (ex: finding an important clue, or procuring a map of the area to be adventured in) are worth CR minus 1, while events upon which the adventure hangs (ex: Tracking down the BBEG's lair, unmasking the Evil Prime Minister) are worth a full CR. Award experience per the CR values given in the Monster Manual and DMG.

Thus if the party has an average level of 4, then finding an important clue has a CR of (4-1) or CR 3, and is worth 700 xp. If the characters were 3rd level, it would be worth 450 xp.

The nice thing about this is that tying the CR for the event to the party's level will tend to scale it with the characters, so that they never lose the incentive to find non-combat solutions, or to forgo non-violent interactions with the environment. Note that my values work for my campaign. Adjust and season for taste.

One last thing, if you do adopt an approach that rewards non-combat interaction, make sure the players are aware that you are doing so. If you don't tell them, they have no incentive to try alternative behaviors.