I’m thinking about building a fixie and was wondering what are the best types of frame, wheels etc to use and how long the whole building process may take?
There's no such thing as a 'best' bike or bike part— most just optimize for different things. Even cheap parts are optimized for low cost. You'll need to give more information on what you want to use the bike for.
Building a fixie could take a week if you go to your local bike store, tell them you have no budget limit, and ask them to order you some parts and build them up.
Or if you slowly accumulate perfect matching vintage parts, it could take a year or more.
Note that if you're optimizing for cost, since 2008 or so, it's pretty much always been cheaper to buy a pre-built fixie than building one from scratch, or converting a road bike.
As a general rule, we don't go shopping on bicycles.SE, since that information goes out of date quickly.
The first problem with building your own bike is fit. If you don't know what measurements you already need from the frame and stem and handlebars and saddle a priori, its going to be hard to gauge which size is right for you. Your LBS may be able to help you in this area, or if you have a similar type of bike as to the one desired, you may be able to move the measurements to some other part set get close to whats right for you. Picking parts is hard enough as it is when things are approximately known already!
The second problem is part selection - brands, budget, functional differences and if they go together (e.g. do you want a wheel with a flip flop hub, or just a fixed gear? What gear ratio should you pick? Do I shell out for the Chris King headset instead of the cheaper Cane Creek one? Will my crank go together with the bottom bracket I selected? etc.) For example, if you have a bottom bracket which is too narrow, you may get some knee pain. Specific to the case of fixies, Sheldon has some advice on selecting gearing and converting old frames to fixies. Also, your frame should have horizontal dropouts.
Finally, theres the problem of putting things together. You may need some help here, as you may not have all the tools on hand for it and the investment in these specialized tools may not be worth it. For example, you may need a bottom bracket tool, cassette tool, chain whip, cable housing cutters (If you run a brake, you can get away with rougher cutting than for shifter cable housing (which fixies don't have), but its still a PITA to cut without the proper tools) etc. Also, if you need to cut your fork at home, its a lot nicer when done by the bike shop instead of pulling out a hacksaw at your house since they typically have nice guides for cutting the fork (with a hacksaw). Sometimes you can borrow these tools from a bike shop or a bike co-op and save yourself some of the expense. Also, you need some expertise to put the bike together in the first place (i.e. some knowledge on how the parts go together and how to adjust them properly, such as chain tension, which is critical on a fixie). Also, you'll make some mistakes (hopefully not forgetting to grease some thing or cross threading things), so you also need to allot time to deal with that. Even once you get the parts put in, you have to tune it to your riding style, so it could be a decent amount of time to put the thing together.
If you are interested in doing this still (instead of buying a bike which you essentially like, and tweaking it, or having a conversion of an old frame done by your LBS), I suggest you go to your LBS, talk to them about what you want to do, take a bike which is similar to what you want, and get them to help you select which parts work for you (and you can have them install things which you don't want to buy the tools for or don't have access to, and have them check the wheel tensioning and stuff). The amount of time it takes you depends generally on mechanical aptitude, access to tools and luck in how close you get to getting something close to right in the first place.
Sheldon Brown's site is useful if you want to do this, as is the Park Tool site (Park Tool makes some excellent bicycle specific tools, but if you're purchasing tools, you may want to go with a cheaper set since they do not need to be so long lasting). There are also various books, such as "The Bicycling Guide to Complete Bicycle Maintenance and Repair" by Todd Downs, "Big Blue Book of Bicycle Repair" by Park Tool and "Sutherland's Handbook for Bicycle Mechanics" among others which may be of use (the last one is designed pretty much for shop mechanics though).
Finally, a note on money: If you want to do this, note that you'll probably end up paying more than a similar bike on the market due to the fact that you're buying 1 of each part rather than a ton of each part like a complete manufacturer does. And you have the cost of your labor or LBS labor in putting it together.
Finally, a note on safety: Despite it not being "cool", I firmly believe fixie should have front and rear brakes (or at least a front brake at the least).