How to make an Apple Script use its local directory?
This is how I'd handle a scenario such as you've described.
I'd place the execute.jar file in the Resources folder within the application bundle, e.g.:
../Untitled.app/Contents/Resources/

Then use the following example AppleScript code within the AppleScript script that is a part of the AppleScript application bundle:
set pathToJAR to ¬
    quoted form of ¬
    POSIX path of ¬
    (path to resource &quot;execute.jar&quot;)

set shellCMD to ¬
    {&quot;/usr/bin/java -jar &quot;, pathToJAR} ¬
        as string

do shell script shellCMD

Now no matter where the application bundle is saved to or copied/moved to, as coded, it will ascertain its location to the resource, the execute.jar file, and be able to run a do shell script command that will execute the shell command accordingly.

To address your comment:

I have an image that stays next to the .jar and is supposed to open with it. If I manually open the contents of the application bundle and run the .jar, the image is loaded in; if the AppleScript code is run, the image is not found. Any ideas?

Try the following example AppleScript code instead:
set pathToMe to ¬
    quoted form of ¬
    POSIX path of ¬
    (path to me)

set shellCMD to ¬
    {&quot;cd &quot;, pathToMe, ¬
        &quot;Contents/Resources/; &quot;, ¬
        &quot;/usr/bin/java -jar execute.jar&quot;} ¬
        as string

do shell script shellCMD