Morrigan was born as the daughter of a mysterious witch, named Flemeth, in the wilds of Cyrodiil. From the time she could walk, her mother trained her in the magic arts. She learned to convert her very life force into extra mana to cast powerful spells. Unknown to the child, her mother had also passed along a simultaneous gift and curse: the affliction of lycanthropy. Flemeth revealed her werewolf nature to Morrigan on her 10th birthday. With help, Morrigan mastered the power of transformation and was soon able to maintain a large amount of self-control as a werewolf. But while her mother was content to remain in the wilds of Cyrodiil, Morrigan was more adventurous, and frequently ventured to the outskirts of towns in wolf form to spy on villagers. Flemeth warned her never to be caught, but as Morrigan repeated the process more and more, she became more and more careless. One day, as she watched a small girl play with her father at a mill, she was found by two wandering members of the Dawnguard. Frightened, Morrigan lashed out against the soldiers, severely wounding both of them. Hurrying back to her mother, she relayed the news, and Flemeth ordered Morrigan to bring the Dawnguard agent back to their hut. Over the next week, Flemeth and Morrigan nursed the two men back to health, and they finally awoke after being completely healed by Flemeth's magic. Initially terrified, they calmed down quickly after Flemeth explained the situation. Sensing her daughter's lust for adventure was still not sated, Flemeth proposed an offer: Morrigan would accompany the Dawnguard back to the province of Skyrim in exchange for the location of a powerful magical artifact hidden there. The soldiers agreed hesitantly, and Morrigan gathered what few positions she had and left her mother for Skyrim. But disaster struck as the band of three attempted to cross the border, as an Imperial regiment mistook them for Stormcloak agents and apprehended them. Morrigan wanted to fight back, but was talked out of it by the Dawnguard members, who didn't want a war with the Empire. The men were set free after explaining their situation, but the Imperials, unable to fully trust Morrigan since she wasn't a member of the Dawnguard, put her on a prison cart bound for Helgen...
Morrigan is inherently distrusting of anyone she meets, as she isn't used to living in a highly populated area. She believes she doesn't need others to help her master the arcane arts, so she will avoid the College of Winterhold. You will obviously need to join the Companions to become a werewolf for roleplaying reasons, but immediately stop doing their questline after you gain this power. The Dawnguard are your default option for the DLC, because as a werewolf you are the vampires' rivals. Don't go out of your way to help anyone unless it will give you a substantial reward, and don't be afraid to coerce or bribe your way out of tricky situations. Remember to utilize the Werewolf transformation as much as possible to be a shapeshifter like Dragon Age Morrigan, and to change up your play style a bit.
The red Vampire armor is the most visually similar to Morrigan's robes in Dragon Age, and they can be enchanted with Fortify Destruction for extra effect. Use any light gauntlets with a good enchantment, and a Mage's Hood for bonus magicka. As a weapon, Morrigan will wield an elemental staff of your choice in one hand, and a spell in the other. This spell can be Lightning, Fire or Frost depending entirely on your preference (though in Dragon Age cinematics she most commonly uses electricity). You can also use Alteration flesh spells to boost your toughness, and the Equilibrium spell to role-play blood magic. The best spells to use are Ebonyflesh, Lightning Bolt, Chain Lightning, Fireball, Fire Storm (from scrolls), Ice Spike, and the Dismay/Animal Allegiance/Ice Form shouts. Morrigan will also be using her wolf form for tougher fights, so work to level that up. Using the Ring of the Hunt, Ring of Instinct, or Hircine's Ring will significantly boost your Werewolf power, so getting one as early as possible is recommended. A necklace that boosts Magicka, Magicka Regen, or Destruction magic is also a must-have.