SQL Server Upgrading Compatibility Level from 2005 to 2012
However what kind of issues can occur going from 90 to 110? What items can break?


Well this is basically very subjective, while I cannot tell you what could break from the code side I could definitely point you towards article which list down all breaking changes. Please refer to Breaking Database Engine Changes in SQL Server 2012.

With my past experience changing database compatibility level does not makes database totally dysfunctional. The compatibility level only defines how certain features would work in certain compatibility level. To help you again MS has listed down such changes in ALTER DATABASE (Transact-SQL) Compatibility Level. The article is thorough and includes details of all versions of SQL server

Using above 2 links you need to jot down the what could possibly fail in your scenario and fix it beforehand. After this is done the process is easy you run the command

ALTER DATABASE database_name   
SET COMPATIBILITY_LEVEL = 110


Please make sure you run this during off production hours. Now you can ask users to test if something fails and it is really important for it to work run the command again and change compatibility level back to 90. So whole process also includes bit of hit and trial method.