Depending on the state of the portable system you have, if its working, not working or just an empty case, keep in mind that it could be worth money on its own. Search for "vintage portable computer" and you'll see lots of systems going for $100-$300.
If you're thinking of doing this because you want put a lot of time into an interesting project, then go for it... but if you're trying to make it practical and inexpensive, you'll probably want to go a more traditional route. Having turned "things" into project PCs before, I can say that it will be more expensive than you think... both in time and in money. Where you'll have the most unexpected expenses will be in the little things... adapters, plates, plugs, tools, etc. And a lot of time will be spent making existing things (keyboard, buttons, switches, screen) do what you want them to do on a modern PC.
For example, an ATX system uses a momentary power switch... an AT uses a toggle switch. You flip the switch and the PC will act like you're holding the power button, so it'll shut back off. Something as simple as replacing the switch may be a pain in the butt.
If you do go ahead with this, I'd love to see it. But I wouldn't blame you for going a simpler route if you just want a small but powerful computer.
Desktop - i5 2500K@4.5Ghz - MSI P67A-G43 - 16GB DDR3-2133 - PNY GTX 970
HTPC - i7 4790 - Asus B85 - 8GB DDR3-1600 - EVGA GTX 1050 TiSelling lots of cool PC stuff on Mercari. Use code VBGQMM for a $10 sign up credit!