I have performed SQL Server upgrade from 2005 to SQL Server 2008. Is there any tools/possibilities to check, whether the upgrade from SQL Server 2005 to SQL Server 2008 has been accomplished successfully?
There are no tools, so to speak. At least not that I am aware. The best thing to do is to test your code, verify your configurations and ensure that SQL Server is functioning optimally. I am hoping that you did this upgrade in a dev or test environment first and that you followed a process with steps that you can repeat again in production. If that's the case, do a full test of your code that interacts with SQL Server. Ensure everything functions as it did before and make sure that all of your maintenance works and that everyone can connect to it at as before.
I would also look in your SQL Server Error Logs and see if there are any issues. Same thing with your windows event logs (particularly your system and application event log)
I recently gave an answer here that talks a whole lot more about SQL Server Upgrades. Lastly, have you considered what you'll do about DB Compatibility Level? If you haven't, you might want to think about changing that for your databases to SQL 2008 and seeing if all works after a regression test. But please make sure you are in a dev or test. I do not advise just changing this in production, nor do I advise starting with an upgrade in production.