What permissions are needed to view dependencies?
The 2012 Books Online topic "Get Information About a View" states that the permissions required for this specific task are:


VIEW DEFINITION permission on the database; and
SELECT permission on sys.sql_expression_dependencies


Note that database-level VIEW DEFINITION is required to allow the user to see information in sys.sql_expression_dependencies; object-level VIEW DEFINITION and SELECT on the DMV will not work (you will receive an empty result set).

Example:

USE Sandpit;
GO
CREATE VIEW dbo.V1 AS SELECT 1 AS const;
GO
CREATE VIEW dbo.V2 AS SELECT v.const FROM dbo.V1 AS v;
GO
CREATE USER Bob WITHOUT LOGIN;

-- These two permissions are not sufficent
GRANT VIEW DEFINITION ON OBJECT::V1 TO Bob;
GRANT VIEW DEFINITION ON OBJECT::V2 TO Bob;

-- This one permission is enough (covers the two above)
GRANT VIEW DEFINITION ON DATABASE::Sandpit TO Bob;

-- Also required
GRANT SELECT ON OBJECT::sys.sql_expression_dependencies TO Bob;
GO
EXECUTE AS USER = 'Bob';
GO
SELECT OBJECT_DEFINITION(OBJECT_ID(N'V1', N'V'));
SELECT OBJECT_DEFINITION(OBJECT_ID(N'V2', N'V'));

-- Show DMV info for the test views
SELECT
    referencing_object_name = OBJECT_NAME(sed.referencing_id),
    sed.referencing_class_desc,
    sed.is_schema_bound_reference,
    sed.referenced_class_desc,
    sed.referenced_schema_name,
    sed.referenced_entity_name
FROM sys.sql_expression_dependencies AS sed
WHERE
    sed.referenced_id = OBJECT_ID(N'dbo.V1', N'V');
GO
REVERT;
GO
DROP USER Bob;
DROP VIEW dbo.V1, dbo.V2;


Regarding sp_depends: that requires only membership of the public role but is deprecated:


  This feature will be removed in a future version of Microsoft SQL Server. Avoid using this feature in new development work, and plan to modify applications that currently use this feature. Use sys.dm_sql_referencing_entities and sys.dm_sql_referenced_entities instead.