Well, think about it this way.
How many watts on your power supply? On my dad's 513C, it's like 250W (or something close to that). That nrrows down your upgrade path. Getting a new CPU= most likely more power = your PSU might not be able to handle it.
Upgrading that rig to a higher-end celeron really won't get you anywhere. I'd advise you to go with a different processor. I'm not sure if your celeron is socket 478 (sorry, don't really know much about celerons). I'm sure processors with the sockets you have are probably not being manufacturered anymore, OR, the highest-end one is probably too slow for today's requirements. So you'd have to get a new motherboard
Fine, even if you DO get a new motherboard...99.99% of low-end HPs are microATX. You'd be hard-pressed to find a good mATX motherboard (you COULD, I'm just saying it may not be 'good'.... <--- don't take that comment too seriosuly though). So if you found an ATX motherboard, you'd need a whole new case.
Also, I'm somewhat confident that you're running out of harddrive space as well... ? MAYBE you'd need a new hard drive. Maybe.
I think your graphicscard/soundcard is all integrated, if you didn't upgrade.
And, like others have said as well, new RAM.
So, let's see. That leaves you to recycle these 3 parts:
-CD Drive/Burner
-Monitor
-Speakers.
I think you're better off building a new computer.
On the positive note: If you wish to simply get a new Celeron 1.3 and not have to worry about making your computer faster, then you won't really have to basically build a new PC, just a thought.
Edit: Why is my previous post blank?
-TPG