She wedded Turgon and then had a child Idril, then after the Kinslaying of Alqualonde she went with him and the rest of the host of Fingolfin crossing the Grind Ice, but somewhere along the way she died.
How did Elenwe die?
Unclear, but she probably dies of cold or exposure during the bitter and dangerous journey.
From The Silmarillion, Quenta Silmarillion, "Of the Flight of the Noldor":
The fire of their hearts was young, and led by Fingolfin and his sons, and by Finrod and Galadriel, they dared to pass into the bitterest North; and finding no other way they endured at last the terror of the Helcaraxë and the cruel hills of ice. Few of the deeds of the Noldor thereafter surpassed that desperate crossing in hardihood or woe. There Elenwë the wife of Turgon was lost, and many others perished also; and it was with a lessened host that Fingolfin set foot at last upon the Outer Lands.
From The Silmarillion, Quenta Silmarillion, "Of Maeglin":
All these things he laid to heart, but most of all that which he heard of Turgon, and that he had no heir; for Elenwë his wife perished in the crossing of the Helcaraxë, and his daughter Idril Celebrindal was his only child.
Those seem to be the only mentions of Elenwe's death, and the issue is not brought up again in Tolkien's Letters or elsewhere.