Solution 1:

Try this:

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

Solution 2:

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);

Solution 3:

I make my own extended class to see what I need, so when I need into my controller or my View, I only add the using to my namespace something like this:

public static class UserExtended
{
    public static string GetFullName(this IPrincipal user)
    {
        var claim = ((ClaimsIdentity)user.Identity).FindFirst(ClaimTypes.Name);
        return claim == null ? null : claim.Value;
    }
    public static string GetAddress(this IPrincipal user)
    {
        var claim = ((ClaimsIdentity)user.Identity).FindFirst(ClaimTypes.StreetAddress);
        return claim == null ? null : claim.Value;
    }
    public ....
    {
      .....
    }
}

In my controller:

using XXX.CodeHelpers.Extended;

var claimAddress = User.GetAddress();

In my razor:

@using DinexWebSeller.CodeHelpers.Extended;

@User.GetFullName()

Solution 4:

This is an alternative if you don't want to use claims all the time. Take a look at this tutorial by Ben Foster.

public class AppUser : ClaimsPrincipal
{
    public AppUser(ClaimsPrincipal principal)
        : base(principal)
    {
    }

    public string Name
    {
        get
        {
            return this.FindFirst(ClaimTypes.Name).Value;
        } 
    }

}

Then you can add a base controller.

public abstract class AppController : Controller
{       
    public AppUser CurrentUser
    {
        get
        {
            return new AppUser(this.User as ClaimsPrincipal);
        }
    }
}

In you controller, you would do:

public class HomeController : AppController
{
    public ActionResult Index()
    {
        ViewBag.Name = CurrentUser.Name;
        return View();
    }
}