If view A is built atop other views and tables, does a user need permission to only view A, or everything down?
Do you need to grant permissions to every object built underneath the
  views? Or just the view itself? Or either one?


Permissions need to be granted to the person executing the query for every object referenced by the view. Except if they are owned by the view owner. In that eventuality Ownership Chains come into play. 


  When an object is accessed through a chain, SQL Server first compares
  the owner of the object to the owner of the calling object. This is
  the previous link in the chain. If both objects have the same owner,
  permissions on the referenced object are not evaluated.


The below creates two tables. T1 (owned by TestUser1) and T2 (owned by TestUser2) and a view that references both tables and is owned by TestUser1.

A third user is granted SELECT permissions on the view but none of the underlying objects. Selecting from the view initially fails for them. However it succeeds after they are granted select permissions on T2. There is no need to grant them permissions on T1 as this is owned by the same user as the view.

Setup

CREATE USER TestUser1 WITHOUT LOGIN;
CREATE USER TestUser2 WITHOUT LOGIN;
CREATE USER TestUser3 WITHOUT LOGIN;

CREATE TABLE dbo.T1(C INT);

ALTER AUTHORIZATION ON dbo.T1 TO TestUser1; 


CREATE TABLE dbo.T2(C INT);

ALTER AUTHORIZATION ON dbo.T2 TO TestUser2; 

GO

CREATE VIEW dbo.V
AS
SELECT T1.C
FROM dbo.T1 CROSS JOIN dbo.T2

GO

ALTER AUTHORIZATION ON dbo.V TO TestUser1;

GRANT SELECT ON dbo.V TO TestUser3;


Test 1 (Fails)

EXECUTE AS USER='TestUser3';

SELECT *
FROM dbo.V;

REVERT;



  The SELECT permission was denied on the object 'T2', database 'Foo',
  schema 'dbo'.


Test 2 (Succeeds)

GRANT SELECT ON dbo.T2 TO TestUser3    

EXECUTE AS USER='TestUser3';

SELECT *
FROM dbo.V;

REVERT;


Cleanup

DROP TABLE dbo.T1, dbo.T2
DROP VIEW dbo.V
DROP USER TestUser1
DROP USER TestUser2
DROP USER TestUser3