This happens to be my particular area of expertise! ;) I have built several bikes buying part by part achieving competitive machines at much as half the price of the competitor! But let's start by not misleading anyone, there is no such thing as a free meal, everything needs a compromise. In my case, I sacrificed looks, trends and time to finally obtain my sweet rides, but it has totally worth it.
First, you have to define a strategy, so, decide why you want a custom built bike. In this case you have stated that you have budget limitations, but for someone else it could be performance, medical or physiological reasons, vanity, etc...
Since your motivation is budget (the same as mine) you have to decide what compromises you are willing to make. Here are my tips:
Every year the component makers release the latest gadgets, and they are always the best performers ever known. Stop there... actually, the performance upgrade from one year component to the next, is almost always negligible, so you are good with last year's derailleur, for example, or even older. This is a great way of obtaining brand new good and cheap (kinda) components, since they are not fashionable anymore their prices drop, sometimes very significantly. You can even get them from perfectly reputable online sellers, you just have to accept that you wont have the latest toys.
It is great to unpack a shiny, brand new component and install it into your bike, but the "new smell" wont make you any faster/stronger (lets face it, there is a placebo effect). There is a reliable way of obtaining good parts by buying OEM (Original equipment manufacturer) parts. OEM parts are components that come in brand new bikes, but they are removed for some reason, for example, maybe the buyer of the bike wanted something else and the shop sells the unpacked, never used item that was left. You will get a perfectly working, branded component, with "marks of installation" but otherwise new piece. You won't get original packaging nor instruction manuals, but for most reputable brands they're obtainable online. (At least Shimano user and installation manuals are available online). Besides, if you really know what you are doing, you do not need all of them.
You don't need top performing parts everywhere (you are on a budget, aren't you?) so you can pick where you want the best performing parts. For example, in my DH bike, I wanted good suspension and brakes, but could sacrifice drive train, so I went with a reputable frame and fork, reputable brakes but choose 8 speed drive train with mixed parts from reputable brands to unheard good enough brands that had available at the moment.
You don't need top of the line parts either. For instance, Shimano has Deore, Deore LX, Deore XT and Saint lines of components (Excuse me if I'm forgetting other MTB product lines). Each has its performance level, but even the least expensive of these, is very good. (personally I use mostly Deore). Take into account why one is better than other. Sometimes it is because of weight savings and in order to make lighter parts they are often made of more expensive materials and with more complicated processes, the price difference can be very significant compared to the performance upgrade obtained. But if what you want is to practice, to simply ride, i.e. you are not racing (for money) you are perfectly fine with the (little) heavier options, or the not so polished ones. Remember top of the line parts are often used by professional riders, who get paid for winning... on the other hand, you and me pay for riding.
Depending on what you are riding, you may be fine with just 8x3 speed drive train (8 cogs in the rear, 3 chainrings in the front). The price difference going from 8 to 9 cogs can be quite big, and you have to change the shifters also. Now there is even 10 cogs for mountain bike, which can be even more expensive.
The other sacrifice you can do is time. If you don't expect to obtain your components right away, you can wait until you find a good sale on the item you want. You will have to work constantly watching prices on line and on local stores. Meanwhile you keep saving money so you gradually increase your budget.
Finally, you can also buy only the frame and keep as much as you can from your current bike. When you get the new frame, check which parts fit and decide which ones you will keep and which ones you have to change/want to upgrade. The shopping list for components just got a lot shorter. Usually you can keep fork, stem, handlebar, grips, shifters, derailleurs, bottom bracket, pedals and tires. Crankset, cassette and chain should be inspected to determine if they are still usable. If they still have some usable life, you can safely shift them to the new bike, and remember to change them in the same date you would if still in the previous bike.
In your case you mention disk brakes, so these should be in your shopping list, along with either a disk compatible wheelset (easier way) or disk compatible hubs and spokes(to re build the wheelset on your own or take it to a bike shop). If you choose cable actuated disk brakes, you can keep the levers, since linear pull (a.k.a. V-Brakes) brake levers are compatible. Otherwhise you need hydraulic levers and hoses.
The final step, after assembling the new bike, is to sell the remnant parts from the old one, thus recovering part of the budget.
This way you can delay your spending, which can be easier on your wallet or may allow you to gradually increase your budget so you can upgrade components as needed.
As previously said, everything is a sacrifice. You may not get a magazine cover worthy bike right away, but if your patience allows it, you'll end up with a bike that is unique, comfortable, excellent performer and you will feel perfectly fine about riding it! :)
