Source 11gR2 database is around 250Gb in size, please suggest me How can I improve the speed of Rman backup.
Pl elaborate on what you mean - how long is the backup taking ? I would suggest you post this as a separate question in the RMAN forum - https://forums.oracle.com/community/developer/english/oracle_database/high_availability/recovery_manager_%28rman%29_2
Yes you can perform a upgrade using RMAN backup and you may increase the speed by using the parallel option in RMAN (assuming you have multiple cores)
But why don't you go with DBUA, I would recommend you to go ahead via this tool, and further more this is what Oracle recommends.