Full backup before and after checkdb
There is no need to take a FULL backup before and after running DBCC CHECKDB. It is recommended that your FULL backup go before the DBCC CHECKDB so that you know exactly which FULL backups don't contain database corruption if DBCC CHECKDB ever fails due to corruption. The FULL backup before the last successful DBCC CHECKDB is one without corruption. 

To recover from corruption, you can use the FULL+DIFF+LOG chain or FULL+LOG chain. Pick the FULL backup from before the last successful DBCC CHECKDB. If there was a DIFF before the last successful DBCC CHECKDB, then you can use that too. But a DIFF after the last successful DBCC CHECKDB could have database corruption, depends when the DIFF occurred and when the corruption occurred.

LOG backups do not contain data pages, so they do not have the corrupted data pages in them. Thus, you can recover with no data loss if you have the entire LOG chain and the tail of the log after the non-corrupted FULL or non-corrupted FULL+non-corrupted DIFF.