Overview
Javascript injection (or Javascript source code injection) is closely related to HTML injection and Cross site scripting. In all these cases the input is interpreted as code rather than data. The code may break out of the current context and switch to another context. Alternatively an injection may stay in the current context but modify the source code.
Video Tutorials
"Context" is the type of code and object into which the injection occurs. For Javascript injection, this is often the value of a Javascript string being dynamically set.
An example of breaking context is injecting script tags into HTML output. The developer believes the context should be HTML (perhaps a table), but the input of script tags (with embedded script) causes the browser to stop processing HTML and switch to processing script. The context switch occurs when the browser stops executing the HTML instructions and instead executes the JS.
Injection within context could be injecting HTML into HTML output. Although the page source code is altered, the context remains the same.
Discovery Methodology
To find the injection, a canary can be used to check for two conditions. The first condition is user supplied or user controlled input is incorporated into Javascript. The second condition is that input is not encoded when output.
Identify the possible input parameters. Does the page take input from a form, URL parameter, cookie, or other input? HTTPFox and Burp Suite are a good tools to see all the input as you "GET" a page (the request) and all the output with which the server responds (the response).
Once you find the page input, try injecting a simple canary like "CANARY-INPUT-1" then search the resulting page to see where the canary showed up. If the input is used to create a portion of the Javascript, it may be possible to break out of the context and inject user-controlled Javascript. To test if the output is encoded, inject a test string such as "CANARY123!@#$%^&*()_+-=[]{};':",./<>?".
Searching for a canary on the actual browser output is not a good idea. Use the browsers "view source" to see the "real" response. Tools like HTTPFox are great for this as well. Tools with more features like Burp are even better but have more of a learning curve. Burp will remember the source of each page you visit as you spider the site.
Exploitation
Once the canary(ies) is located, identify what characters need to be injected to "end" the current instruction. Identify the characters that are needed to block out any instruction that comes after the canary. Put your injection in the middle.
Example
The Password Generator page allows JS injection. The JS that builds the password is a diversion. The injection point is elsewhere.
To help see if your Javascript injection forms valid syntax when incorporated with the sites Javascript, try validating the resulting Javascript at an online validator like http://www.jslint.com/
An injection that works on the Password generator page is
canary";}catch(e){}alert(1);try{a="
Videos
Cross-Site Scripting Explained - Part 8: Javascript String Injection