Manual testing is awesome, but automated checks are faster and maybe even more reliable. Here are some self-written tools you could use for your purposes:

#1
Joomla QPersonel Exploit
http://www.xenuser.org/exploits/joomla_com_qpersonel_sploit.py

#2
Automated Joomla SQL Injection Exploiterhttp://www.xenuser.org/exploits/joomla_sqli_sploiter.py

#3
Joomla BF Quiz Exploithttp://www.xenuser.org/exploits/joomla_com_bfquiz_sploit.py

#4
Column Fuzzerhttp://xenuser.org/tools/column_finder.py

#5
Simple SQL Injection Vulnerability Scannerhttp://www.xenuser.org/tools/sqli_scanner.py

#6
Simple Log File Analyzerhttp://www.xenuser.org/tools/scan_log.py

#7
Simple Local File Inclusion Exploiterhttp://www.xenuser.org/tools/lfi_sploiter.py

These tools can help you to exploit vulnerabilities within Joomla or some extensions.

NOTE: Read the help (included in those tools) for details.