Not much is known about Kurenai's history prior to the manga.
What is known is that she is the #1 Genjutsu specialist in the village, and at the time of the manga's opening that she is newly appointed to her rank. She is an Elite Jounin in rank: the highest rank that can be achieved short of Hokage. She is one of only four Elite jounin displayed in the village along with Gai, Kakashi, and Asuma. Her skill in Genjutsu has been stated by Kishimoto in the databooks as being 'equal to Itachi', however despite matching him in this area, she falls behind him in all others. She is a strong willed and confident woman and her looks are renowned. She tends to dress in red and white clothing that and has a distinctive style that is likely reminiscent of the Japanese Geisha, a lady who lives their life by grace and tradition. However this means she lacks the body armour favoured by Asuma and Kakashi and thus must take more care in close combat.
Kurenai is the leader of team 8 whom are tracking specialists from elite ninja clans, an odd choice of leader until you realise how well Genjutsu and stealth synergises with intelligence gathering. She is well respected by her peers but has not shown great strength in battle, being more of a thinker and strategist, relying on illusion and sneak attack over brute strength. This is complimented by her lover Asuma, whom almost always fights by her side in battle. When fighting beside Asuma all the bases are covered, strength and close combat skill with long range illusions and disabling abilities.
Of all her students, Kurenai is shown to be closest with Hinata, whom she treats like her own child. She constantly strives to build Hinata's confidence in herself, as Hinata has a strained relationship with her real parents and clan.
Kurenai is shown to be very close with Asuma, and she was often shown worrying about him when he was on active duty. In battle they complimented each other in combat perfectly. When questioned about their relationship they always tried to avoid the subject, showing that as working colleagues they were both concerned about how it might harm their careers of jeopardise their missions. However their relationship is something of a joke amongst the villagers as almost everyone has heard about it.
In recent chapters, Kurenai is shown pregnant with Asuma's child and therefore is not on active duty. This clearly shows that both Asuma and Kurenai were pursuing a relationship in secret.
Kurenai is shown to be a master of Genjutsu: mind magic that affects the brains and nervous systems of others. She can use this to manipulate opponents perceptions or movements, making them act and behave erratically. She can change the very world around them in their perceptions, and can even physically harm an opponent: either by rending herself and the environment invisible and attacking, or using it to make the opponents bodies believe they are being harmed or bound at which point the body reacts accordingly.
Kurenai is also able to utilise standard Shinobi techniques to a far greater level than most as an elite jounin. Therefore she is super fast and accurate, able to leap great distances and avoid attacks with superior agility.