You can identify costly queries (and the databases they are associated with) using the DMVs, e.g. from this TechNet article:
SELECT TOP 50
[Average CPU used] = total_worker_time / qs.execution_count,
[Total CPU used] = total_worker_time,
[Execution count] = qs.execution_count,
[Individual Query] = SUBSTRING (qt.text,qs.statement_start_offset/2,
(CASE WHEN qs.statement_end_offset = -1
THEN LEN(CONVERT(NVARCHAR(MAX), qt.text)) * 2
ELSE qs.statement_end_offset END -
qs.statement_start_offset)/2)
,[Parent Query] = qt.text
,DatabaseName = DB_NAME(qt.dbid)
FROM sys.dm_exec_query_stats qs
CROSS APPLY sys.dm_exec_sql_text(qs.sql_handle) as qt
ORDER BY [Average CPU used] DESC;
These will tell you about heavy hitter queries, but unfortunately it won't pinpoint a database that might have very high volume of small queries that are using small bits of CPU individually but large bits in aggregate. You can do that with this query from Glenn Allan Berry's DMV queries:
WITH DB_CPU_Stats
AS
(SELECT DatabaseID, DB_Name(DatabaseID) AS [DatabaseName],
SUM(total_worker_time) AS [CPU_Time_Ms]
FROM sys.dm_exec_query_stats AS qs
CROSS APPLY (SELECT CONVERT(int, value) AS [DatabaseID]
FROM sys.dm_exec_plan_attributes(qs.plan_handle)
WHERE attribute = N'dbid') AS F_DB
GROUP BY DatabaseID)
SELECT ROW_NUMBER() OVER(ORDER BY [CPU_Time_Ms] DESC) AS [row_num],
DatabaseName, [CPU_Time_Ms],
CAST([CPU_Time_Ms] * 1.0 / SUM([CPU_Time_Ms])
OVER() * 100.0 AS DECIMAL(5, 2)) AS [CPUPercent]
FROM DB_CPU_Stats
WHERE DatabaseID > 4 -- system databases
AND DatabaseID <> 32767 -- ResourceDB
ORDER BY row_num OPTION (RECOMPILE);
Neither of these queries identifies the application that ran them, and the DMVs used don't store that information (you would have to catch the queries in the act and note the application name in sys.dm_exec_sessions, or review a trace).
Of course you can automate this work with a variety of 3rd party performance tools on the market (disclaimer: I work for one of them, SQL Sentry, who produces Performance Advisor, which does all of the above, including keeping track of high-cost queries and maintaining the information about which database they ran in and what application called them).