You can't upgrade a laptop
Not true. You can upgrade the hard drive and memory on almost any laptop, and the CPU is upgradable (though not necessarily easily) on most. Some allow for the GPU being upgraded as well.
Not going to happen, even top of the line desktop cards with fast processors struggle with that game on max settings.
Like other posters have mentioned though, you probably can't upgrade anything in that computer other than the memory and possibly the cpu.
I've run the Company of Heroes demo on max settings on my laptop (GeForce 8600M GT DDR2) with decent FPS. It wasn't 60 in all likelihood, but it was certainly smooth enough for an RTS.
The processor is probably nonreplaceable and it certainly doesn't have an empty slot for a video card
I'm not an expert on AMD processors, but I know mobile Intel ones are almost always replaceable (though it's not an easy upgrade, and you can't upgrade to just any other Intel processor). As for the video card, it probably isn't replaceable - most are soldered on in laptops, and many others require a specific connection (thus requiring you to purchase the new card from your laptop manufacturer). However, if your laptop is MXM-compatible, it can be upgraded without too much difficulty. Otherwise, it depends on your laptop manufacturer.
If your graphics card is the GeForce 6150 or another integrated card, then you probably don't have a dedicated graphics card slot, and don't have any chance of upgrading. But I'm guessing that since you said it's a media-center laptop and has a 6*** series card that you have the 6600 or another dedicated card.
If you have a fair sum of cash, keep the media center a media center and buy yourself a nice gaming desktop. A decent one that will last you a year or two shouldnt cost more than 1000-1200$ US
This is probably what you want to do if you want betting graphics performance. Here's what I'd advise:
1). If you want to upgrade: Start with memory. 1 GB really isn't enough for gaming with Vista. I'd buy a 2 GB stick so you'd have 2.5 GB total. Total cost will be about $60. Then graphics card if possible (depends on the model, but it's unlikely it's possible). Then processor - AMD Turion X2 Tl-68 (2.4 GHz) is the top-of-the-line on the AMD side, which you are limited to by your motherboard.
2). If you can't get what you want by upgrading (i.e. can't upgrade video card): Look for something with at least an 8600M GT or 7900 GS from nVIDIA, or 2600 from ATI. The Gateway P-6831FX is probably your best bet overall for price and performance, selling for $1350 at Best Buy with the high-end 8800M GTS video card, but it is a 17-inch model.