Lanfear is portrayed as an envious, manipulative, narcissistic, self-obsessed megalomaniac with extreme impulse control issues. Basically, a psychopath with Borderline Personality Disorder, tremendous beauty and with a huge amount of power.

She goes batshit crazy because Lews Therin denies her the Dragon dick and picks Ilyena after the relationship Lews Therin has with Lanfear goes bad because she's vain and really only wants to be with him for the prestige of being with the [literal] most powerful man alive.

Worth noting that as with Lews Therin Telamon, Ilyena also had three names, Ilyena Moerelle Dalisar. A third name is a point of prestige in the Age of Legends granted as a mark of profound respect and achievement. Lanfear's full name? Mierin Eronaile. No third name because for all her looks she isn't actually that impressive beyond her appearance and raw strength in channeling which was the maximum for women but still vastly underneath what men could reach.

https://wot.fandom.com/wiki/One_Power#Scales_of_Power

But even though she was as strong in the Power as a woman could be, Mierin did not distinguish herself enough to earn a third name; a mark of great social service and prestige that she coveted. This presumably irked her, as she was exceedingly power-hungry. Lews Therin himself eventually left her because she loved the prestige of being his lover more than she loved Lews Therin himself.

She was known to have been quite bitter over Lews Therin leaving her, and made several blatant and public attempts to get him back from his wife Ilyena Therin Moerelle. Lews Therin, however, always rebuffed her advances, as he loved his wife and knew Mierin's character too well.

https://wot.fandom.com/wiki/Lanfear