Hmmm well you obviously never bothered much with the Federal Navy missions...
Well I recommend that you trade in the Panther for an Asp Explorer, fit it with all the best electronic warfare from a top tech system like SOL, a Naval drive unit to save on cargo space, 5 or so shields, 20MW beam laser, naval ECM, naval missiles etc, basically kit it out so that you have just enough room left for your fuel and if necessary your passenger cabins.
Then start doing the federal or imperial naval missions (depending where you are in the galaxy), making sure you can actually do them in the time allotted (ignore any of the silly impossible Alpha Centauri missions), as if you are late your ranking will drop.....
Also see if any other missions or passengers are there for the same trip....
As I said, the Asp Explorer is in my opinion the best balance between size and speed making it ideal for dangerous missions.
When you reach the assassination missions you will however need more of a behemoth with an accelerator to carry those out.
UPDATE: (following my waking up)
So you are trying to improve your ELITE rating and not your Federal or Imperial ratings... ok
Your ELITE rating increases the more enemy ships you blow up.
Your FEDERAL or IMPERIAL military rankings increase respectively as you complete, within the alotted time, the missions taken up at systems controlled by respective powers.
The higher your military Rank, the more dangerous will be the missions offered, and greater the rewards.
Your two seperate military rankings are just that, and are not affected by each other.
To improve your ELITE rating, go for the the Asp Explorer again, and check the missions for people wanting "passage on a fast ship" and offering usually over 3000 credits, invariably they will be running from someone and you will encounter some pretty nasty enemies whilst trying to taxi them to their destinations.
Also some systems are inherantly dangerous, simply ferry any cargo to these systems and you will be plagued by attacking ships, in this respect the clippers would be ok but with these its easier to target a ship and fly into it that to shoot it due to the sluggish maneuverability of these fat ships.
The asteroid bug works apparently, but I don't know how..