OK, you have all your parts in a big pile, how do you put them together? Well, as with so many things, the answer to that goes to the adult diaper industry...

Which is to say: "It Depends!"

[Insert chorus of groans here...]

Thank you, thank you, I'll be here till Thursday. Try the veal!

*ahem*

There are rather a lot of possibilities here, honestly, and your best option really does rest on what you have. The central determining factor is where you're getting your mainboard, so I've organized a limited build guide here based on that. Rearrange the steps to suit your individual build, the sky's the limit. Each likely scenario is in bold, so scroll to the one that best fits you, and go from there.

If you have an Intel Compute Stick, a clone, or a compatible as your system unit...

...things are fairly easy. This is your textbook basic build. You should have two USB ports -- one might be USB2.0 and one USB3.0, but it's *highly* unlikely you'll have more than two such ports -- plus a MicroUSB for power, a MicroSD slot, and a male HDMI port. If you're lucky, you'll have a separate audio-out, but usually this is handled by HDMI only.

You should plan on a cheap USB 'sound card', unless you like to argue with Linux, because configuring sound output on these things is typically something of a challenge at best. A USB wireless card is also a good idea -- do your research, though, and try to avoid anything based on Realtek chips. DeviWiki is a good resource, but their ad provider is truly a creation that would scare HP Lovecraft himself; have good adblock software on desktop (AdBlockPlus or UBlock Origin come highly recommended; I can vouch for the former, and my pal who runs the local tech shop prefers the latter) and for the love of all that is good and righteous in this world, do not go there on a phone.

You also *absolutely* have to have a powered USB hub; the circuitry that runs those two ports is far, far too anemic for anything but the most minimal of peripherals. You can run a wired keyboard and a wired mouse, or a wireless combo and a single USB stick, but much of anything more will be entirely too taxing for it. A seven-port hub here will mean that you have more to use than a single port without unplugging things... if you even have that! A four-port hub will only leave you one spare USB connector if you're using a keyboard and mouse combo that takes only a single plug... or if you forgo either the WiFi adapter or USB 'sound card', in which case you will either have a truly awful Internet connection (when you have one at all!) or a mute machine.

Also, expect poor thermal performance, even if you have good ventilation. The early Compute Sticks (and clones/knockoffs) were entirely fanless, and the current ones have only token active cooling. Even a dinky 40mm PC fan goes a long way here... something ripped out of a dead laptop, particularly if it's known for sounding like the sort of jet propelled smoothie machines coffee shops use, is truly invaluable.

Basically, your hookup guide is:

Hang the external hard drive or SSD off one of your two USB ports.
Hang the USB hub off the other USB port.
Run an HDMI cable to the controller board you got for your LCD screen (or whatever other display solution you're using)... you will very likely need a female-to-female gender changer here.
5v power goes to USB hub and to MicroUSB on the stick PC.
12v power goes to the LCD controller PCB.

Once you build your structure, housing, whatever, and plug in your peripherals, you're done.

If you have a MiniPC...

...really it's the same situation as with the Compute Sticks, clones, and compatibles, except that cooling will likely be an even more thorny issue than with them -- the MiniPCs really seem to like having huge passive heatsinks with no fans whatsoever, when they really need fans -- and you will almost certainly have an audio jack (unified for microphone and headphones/speakers) and three or four USB ports.

You also may have a system board that needs 12v or something closer to laptop voltages like 19v. If it's a 19v board, it likely will run on 12v anyways -- give it a try, it's worth it!

Follow the build guide above, making exceptions for the fact that you might not need a USB 'sound card' and can almost certainly get away with a four-port hub here... and, again, you may have to deal with strange input voltages.

If you have a disassembled laptop...

...depending on what you have, and *why* it was disassembled, this can be an advanced project or a simple one. Again, I don't encourage Windows *tablets* because they usually use a strange display protocol called MIPI -- and finding compatible displays that aren't your own is nearly impossible without a considerable degree of reverse engineering followed by custom cable making. This is made even worse by the fact that tablets often leave behind one of the common 'safety' features of laptops, as far as repair and recovery are concerned -- even when they *have* a secondary display output, such as a MiniHDMI port, whether or not it mirrors the primary display on boot by default is anyone's guess. (Laptops have done this, reliably, for *decades* now, as a simple diagnostic aid.)

I also don't recommend Chromebooks -- many of the lowest-end ones are actually based on ARM processors, especially if they're early systems, and even if they aren't, the BIOS has some Google-enforced special sauce inside that effectively keeps you from natively booting any other OSes. (Technical: you have to have an OS kernel with a Big Daddy Google signature -- which is to say, a Chrome OS kernel and nothing else.) You're looking at installing something like coreboot to make that hardware truly useful -- which is not something to sneeze at, even for the experienced! Just don't.

Assuming you have all the parts (except maybe the battery, optical drive if there is one, keyboard, and touchpad) and they all generally work, *and* you can wire the power button in a reasonably coherent way -- probably the best approach would be a new custom housing physically incorporating the power brick into the body. At that point your steps are something kind of like...

Assemble the laptop components in your new case structure. Be sure it can vent heat well!
Add a hard drive on the internal connector if you can, via a USB port (or eSATA if you have one, that's better for that) otherwise.
Hang your powered USB hub off an internal connector.
Hang any additional USB peripherals (eg a USB WiFi adapter if the internal is shot and USB is cheaper/easier, or a USB 'sound card' if the internal sound chip is hosed) off the hub.
Add keyboard and mouse, hang off of USB hub.
Incorporate the lid and screen from the old laptop, in as intact a fashion as you can reasonably manage.
Shuck the case off the power brick, and incorporate it, along with the USB hub power supply.
Wire up a mains-rated master power switch, and wire an inlet connector (the "IEC" connector off an old, dead ATX power supply -- or any of the zillions of other things that use it -- is a good choice) to that switch on one side. Wire the power bricks to the switch as well.

Good luck!

If you have a compact desktop (SFF / USFF desktop, Mini ITX board, NUC box, thin client, etc)... 

...extra credit time! This can get REALLY interesting -- and either really easy or really hard, really fast, depending on how much you know and are prepared for, and how resourceful you want to be.

Mini ITX systems and those with standard ATX/ATX12V (20pin/24pin, respectively) power connectors can use things like the PicoPSU for power as long as you're not driving anything ridiculously beefy. Otherwise, it's down to working out supply connector pinouts and wiring an adapter, or reusing the existing supply somehow. NUCs, some SFF/USFF desktops, and a few Mini ITX boards on the weirder side of things are designed to use external laptop-style power bricks instead.

Thin clients are also a nifty choice. There's a kind bloke in the UK who collects them and has basically a closed-source Wiki chock full of unmistakably priceless information at https://www.parkytowers.me.uk/thin/ -- an invaluable resource. Click "Details" in the tabs at the left side of that page and you'll be dropped clean into the midst of it all :) It's organized by manufacturer (make), then model. He doesn't have every single one on Earth in there but he *does* seem to have the vast majority. This is SUCH a help because these systems are not only incredibly minimal by design, they also are incredibly *weird* -- the whole point is maximum cost reduction vs a standard (ish) PC, after all -- and so you get all sorts of crazy things, from proprietary power supplies, to incredibly out-of-date connectors and ports in odd shapes.

Often, in fact, you'll be dealing with an IDE interface designed to directly connect to a notebook hard drive or equivalent, so it's actually a 44pin IDE male header! SATA connectors are actually quite rare, and mSATA even rarer. (Don't even *try* for M.2/NGFF, you're almost guaranteed complete disappointment.) Where SATA connectors *are* present, you'll be encountering a unified 22pin connector with power and ground both, made for something quite a bit more compact than a standard laptop drive and placed in a way that makes such a drive impossible to use without extension cabling, the vast majority of the time. This is true even of relatively modern such systems.

Also, nearly always these are designed to be fanless systems, and they really suffer in that way. Rigging some sort of active cooling is *highly* desirable here where at all possible.

Proceed as for a laptop or MiniPC, whichever makes more sense, making adjustments as required by the hardware you have.

OK, it's built. Now how do I make it boot...?

That's the next log. See you there...