At the moment, the Big Bad is confirmed to be Cassandra Nova. Buckle up, because her Origin Story is seriously fucking weird, even by X Men standards.
Nova began her life as a Mummudrai. The Shi'ar believe every living being must fight it's Mummudrai - it's polar opposite - on the Astral Plane before they are born. Because Nova was Xavier's Mummudrai, she was able to use the vast potential of his Mutant DNA to construct an actual physical body in the womb. She then attempted to strangle Xavier with his own umbilical cord. Xavier defended himself with his Telepathy, which caused his mother to go into labor.
Nova was believed to have been stillborn, but in fact survived as a, "chaotic mass of cells" in a sewer. Over the decades she slowly grew a human-shaped body. She was also able to access the full range of Xavier's abilities. Not just the powers he did have, such as Telepathy, but also Telekinesis and a Healing Factor, which he potentially could have exhibited. She can change her own DNA; in her debute she replicates the DNA of a nephew of Bolivar Trask, allowing her to take command of Wild Sentinals. She then killed Trask and used the Wild Sentinals to wipe out Genosha, killing 16,000,000 Mutants.
Nova is also a extremely competent tactician, as she can see the future to a certain extent and plan accordingly.
How much of the above will be adapted into the movie is unknown as yet, but she clearly possess high level psychic abilities, given how easily she defeats Wolverine in the trailer.