Hey Spork,
It appears that the face normals of your character are inverted.
You can go about fixing this by either editing the mesh config in UE4, or manually flipping them in blender, the former being the easier method, which is what I will explain.
(IN UNREAL ENGINE)
Open up the static mesh, and in the details panel find and enable “Double-Sided Geometry”
Next, go into the material you have assigned to your character and under Details enable “Two Sided”.
Apply the material and save all, and it should work.
Hope this helps,
-Peem