How to disassemble a laptop

Not knowing this particular model, the generic answer to the question title is "With care and patience". However, as there are as many methods as there are models, an instruction booklet is the best help. Sometimes these can be found on the maker's website; did you try that?

Otherwise, I think you're on the right track: remove all screws, check inside drive bays and under bottom doors for hidden screws. Screws are sometimes hidden under the laptop's rubber feet. Try to get the KB off! Sometimes the panel between KB and screen hinges is easily removable by sliding to one side; underneath may be interesting screws or clips that hold the KB. See if you can get the screen to come off, that may help the thing fall apart.
/RID

If you google the laptop model and something like " replacing motherboard", " opening up", etc. you can usually find detailed instructions on how to do this.

Hi Eugene! All the previous advice is really great. Firstly, don't force anything. Stuff, when done correctly, just slides open. I recently put in WiFi radio in my laptop and i must say that some models require you to remove perriferals before you can open the case. Like with my HP Pavilion, i had to remove DVD/CDR drive (the whole device) before the case would open. You've got to get a manual or some documentation somewhere and read on how it's done for your laptop! Remember, don't force anything at all, or you'll break it, handle it like a new born :)
-=G=-

I couldn't google anything with your Time 8375 description. Is it a MiTAC or Advent 8375 maybe? Look at the back stickers on you lappy, see what it says there. I'm sure you can get some documentation for your machine.
-=G=-

Thanks for the comments.  I have Googled all the combinations of Time, laptop, notebook, motherboard, disassembly, open, 8375  I can think of but come up with nothing.  Indeed an instruction manual would be excellent, but I don't know where to get one.  Time is the only manufacturer's name on the case, although could be rebadged.  TIme was bought by Tiny and they then went bust, so no manufacturer's site to visit.  The best I have found is
http://www.pchub.com/uph/model/1--2742/8375-parts.html
I think I may see the screws which are holding me back, but I need to remove the screen to access them, and having removed all screws from the screen, it just won't give.  It may just be a few plastic lugs holding me back, but if I force the issue I will not have an even less saleable item for eBay.  I can't even find a forum or usenet group which seems like it may be able to help.
<sigh>

The screws for the screen may well be at the back edge of the casing, where the connectors are and not on the screen itself or the hinges. In some cases the screen and hinges will come up out of the casing when the screws are loosened. Take care that the screen cable is not damaged in the process.
/RID

I would also recomend removing your keyboard, a lot of times there are a few screws that are undernieth the keyboard, which hold your case together as well. You want to unscrew them. I do need to take the screen off, otherwise you risk to break something off, because the screen is usually heavy.
-=G=-

Ha.  Statick001, all points and kudos must go to you.  Thanks to be shared equally with all other contributors who made as sensible suggestions as is possible without having seen the particular model.