I'm sorry but if you really want to PVE and PVP on a dragonknight, you have to make changes to both skills, gear, and champion points when you switch between the two in order to be most efficient.

1. Armor Sets
Your armor set should have a different trait for pve and pvp. This is why it is important to have two sets of armor. In PVE, you will find that the most beneficial traits for DPS magicka users are respectively divines and infused. Most people use infused on large armor pieces, meaning helm/chest/legs/shield, and divines on the smaller pieces, shoulders/gloves/waist/boots. However, there are also people who use all divines which is what I would recommend for PVE content as a DPS. For PVP content, as a magicka DK dps it is my recommendation that you run impenetrable on every piece of armor. The damage mitigation it provides is too great to go unrecognized.

Currently on my Magicka DK for PVP I am running....

5L/1M/1H

2/2 Valkyn Skoria (Light Infused helm/ Medium Impenetrable Shoulder both with max magicka glyphs) My preference would for both to be Light but i wasn't that fortunate on my drops.
5/5 Julianos (Heavy Chest/ Light gloves/legs/waist/boots all impenetrable trait all with max magicka glyphs) Again I would prefer heavy chest and medium legs but the monster set drops were not on my side in this department.
5/5 Vicious death (Sword sharpened trait disease glyph/ Shield impenetrable trait max magicka glyph/ Necklace arcane trait magicka recovery glyph/ Ring arcane trait spell damage glyph/ Ring arcane trait spell damage glyph). In the current Meta, you really cannot beat this armor set. It makes taking down groups of players rather easy.

If you do not have access to the monster sets, I would run Magnus Gift on all pieces that are not Julianos, and then get your hands on the Willpower three piece set for necklace, and 2x rings.

2. Weapons and Skills
In regards to weapons, this is up to you. End game magicka DK's will suggest double destruction staff for PVE content. This is because at end game, it is not your job to mitigate damage. Yes you need to stay alive, but this is usually done by staying within distance of the healer in the group, only tossing a self heal every once in a while where needed (i.e. the healer is stunned/disoriented). I would not recommend sword and shield for PVE content as it will decrease your damage significantly compared to a destruction staff. For PVP, this is entirely preference. I have seen DK's who run double sword and shield, sword and shield/restoration staff, sword and shield/destruction staff, and destruction staff/restoration staff. It is entirely preference and many good players will agree with that. It is essentially what works for you in staying alive, while being able to dish out damage. You also need to take into account what type of grouping you will be doing in PVP. Are you the kind of person who runs with 6 friends and goes around mobbing other small groups? Or do you play in a 20+ man zerg group taking/defending keeps? Again, the choice is yours to make.

As for skills. This is also preference. Do you like to play as a support? I hope so... because that's what a DK is all about. In your best gear sets you are still a support class. Sure you can 1v1 and maybe 1v2. Anything more than that gets hairy and being outnumbered puts you at a severe disadvantage unless you are extremely skilled. However there are some skills that I find are beneficial on every type of build. These skills include....

Shooting Star Ultimate (Morph of Meteor). This skill is your go to for burst damage. Combining it with other skills is potentially the best way for you to kill and opponent.
Burning Talons (Morph of Dark Talons). Probably in the top 5 best immobilizing skills in the game. Both stamina and magicka users hate this skill because it forces them to drain A LOT of stamina to stay out of it and it can be used three times before immunity.
Dragonfire Scale (Morph of Reflective Scale). This skill is currently bugged on console, however, it allows you to reflect 4 spells back at your opponent for 35% more damage. This allows you to stay alive through snipe spammers and other ranged classes and is regarded as one of the best skills in the game, period.
Flame Lash (Morph of Lave Whip). This is your spammable attack. When combined with an immobilizing effect like burning talons, its gives a powerful self heal.

As for other skills, this is up to you based on your weapons choices (destro staff/ resto staff/ sword and shield). You want to make sure that you are slotting skills for different purposed. Think about how you attack... are you up close and personal? Or do you attack from ranged? If you are up close and personal then you need to have a gap closer in either empowering chains, or shield charge.

In the end.... Many of the choices are yours to make. I have spent countless hours doing research, testing armors, different skills, as well as spending hundreds of thousands of gold on respecs and armor changes before I found what I liked.

Best of luck!

Hutch