MVC 5 Access Claims Identity User Data

Try this:

[Authorize]
public ActionResult SomeAction()
{
    var identity = (ClaimsIdentity)User.Identity;
    IEnumerable<Claim> claims = identity.Claims;
    ...
}

You can also do this:

//Get the current claims principal
var identity = (ClaimsPrincipal)Thread.CurrentPrincipal;
var claims = identity.Claims;

Update

To provide further explanation as per comments.

If you are creating users within your system as follows:

UserManager<applicationuser> userManager = new UserManager<applicationuser>(new UserStore<applicationuser>(new SecurityContext()));
ClaimsIdentity identity = userManager.CreateIdentity(user, DefaultAuthenticationTypes.ApplicationCookie);

You should automatically have some Claims populated relating to you Identity.

To add customized claims after a user authenticates you can do this as follows:

var user = userManager.Find(userName, password);
identity.AddClaim(new Claim(ClaimTypes.Email, user.Email));

The claims can be read back out as Darin has answered above or as I have.

The claims are persisted when you call below passing the identity in:

AuthenticationManager.SignIn(new AuthenticationProperties() { IsPersistent = persistCookie }, identity);