Alright with the information you've provided I can go into a little more details about recommendations.
So, with this business the general computer tasks are lightweight as well as the types of files that they are going to be accessing to. Budget wise this is a good thing. The types of computers you will need to utilize can generally be less powerful and lower cost systems then, but I would HIGHLY recommend looking into solid business-class computers instead of consumer line computers. This means looking at the HP Business Desktop or ProBook (notebook) lines instead of the Pavilion or G series. Or if you prefer Dell, looking at the Optiplex line instead of the Inspiron. First off, the quality is much better on a business line computer, and second the support and functionality for a business network is much more improved where as the home consumer lines are just not built for or capable for that sort of thing. Also, stick with Windows 7 Professional or Windows 8 Professional due to the ability to have many of the business features (such as joining a domain.)
Given the size of your network you wouldn't technically have to have a full server. You could operate within a workgroup still, which doesn't need a centralized computer to manage, but it does take more work to manage all the sharing and access permissions. You would instead use a NAS, a basic file server designed simply for sharing out files. A simple NAS is going to be cheaper to begin with, but it's not as flexible, it's not upgradable, and it's not going to give you the ability to add more roles into the future (such as domain services.) Basically, if you foresee your business ticking at about ten total computers for the next few years, then you can make do with a NAS, but if you foresee growth within the first few years here, then I'd recommend go ahead and invest a little more to get a full server that CAN do domain services and other features so that you have room to grow.
Network wise, it is always recommended to run wire instead of wireless. First off, its more reliable. A wired connection almost never has data loss or random outages, but a wireless connection is going to suffer from interference and attenuation. Also, a wired connection can be much faster than a wireless connection. A gigabit switch can be very cheap to get into, giving you a full gigabit of throughput to each endpoint device. However wireless routers on average operate with a theoretical maximum of 300 Mbps, which is split across all of the connected devices, and of course degraded speeds occur with more interference and distance from the access point. This means realistically with ten computers running wirelessly you're instead getting about 10 Mbps per device which is going to crawl working with files on a central shared storage device or server. But yes, it can be done, you just have to invest in a nice high quality router or access point built to handle that size of a network load.
Most of your home wireless routers can support up to five wireless devices simultaneously without much problem. But more than that and it just really starts to impact network reliability and performances, they just aren't designed for more than that. I would recommend looking into a nice secure router (such as a Cisco VPN router or Sonicwall TZ 105 firewall appliance) and then run a separate enterprise wireless access point off of that. Think of the access point as just a high quality antenna connected to the router for all of the actual "work" in the network. Enterprise access points are not only designed for handling a greater number of concurrent devices, but also tend to have a better coverage range for better signal strength and thus better reliability and speeds.
A budget of $5K is probably going to be pushing it. For your end workstations, all the necessary peripherals (monitors, keyboards, printers, etc.) and the necessary licensing (Microsoft Office 2013/365, plus anything else you need) I'd suggest budgeting about $1000 per computer. So already that's double what you are talking about. You're also probably going to want about $500 to $1000 for your network (depending upon if you will need to run additional ethernet cabling, or purchasing additional wireless adapters, etc.) And then you still have your storage device or server. If you keep it cheap with a simple NAS you're still probably looking at around $1000 or more for a decent quality device with the storage capacity you need. For a full server system, you're going to at least double that, probably somewhere around $2500.