First, the "Can I Run It?" test is a very inaccurate test, it's often wrong and merely compares the specs it finds for your PC to the requirements for the game. It usually means you'll be able to run the game, whether it'll perform to a playable standard or not is another matter.
From previous experiences, the Celeron chips are really quite bad in terms of performance. I had a 3.2GHz single-core Celeron that didn't perform like a 3.2GHz clocked chip at all, it bogged down at the simplest tasks.
I remember that Team Fortress 2 used to bog down under high-action on my PCs old specs, of 2.2GHz non-OC'd
CPU and a 9500 GT 1GB. I haven't played it with these current rig specs though.
A 9400 GT isn't a great card, it's pretty much a budget solution and you won't get much out of it. Plus, the PC is a Dell, and even basic upgrades won't hide the fact it's a workstation rather then a gaming, or even multimedia PC.
If your not already, try running TF2 at it's very lowest quality settings. If it plays OK then, you could try slowly improving the settings to find your optimum quality/performance ratio.
Vger also bought up a good point, that
PSU will be a generic brand used by Dell and designed to provide the bare minimum power for the original system to work off. Adding a new GPU could be asking too much of such a generic component.
The only other thing I can suggest is getting some extra cooling. You mentioned that you have no fans or coolers? You must obviously have a
CPU cooler and the GPU should have a built-in heatsink/fan. But adding some cheap case fans could provide a better airflow through your case and reduce component temperatures. Sometimes the stress of running games can send component temperatures quite high and this
can cause some stability issues.