This is about as cheap as you can go with using new parts.
You can build with used parts (allowing to reuse your RAM as well) but, this is all new.
PCPartPicker part list / Price breakdown by merchant
| Type | Item | Price |
|---|---|---|
| CPU | AMD A6-7400K 3.5GHz Dual-Core Processor | $60.99 @ NCIX US |
| Motherboard | *Gigabyte GA-F2A68HM-HD2 Micro ATX FM2+ Motherboard | $43.98 @ OutletPC |
| Memory | *Kingston Savage 4GB (1 x 4GB) DDR3-1866 Memory | $30.99 @ Amazon |
| Storage | A-Data Premier Pro SP600 32GB 2.5" Solid State Drive | $32.99 @ Amazon |
| Case | Cougar Spike MicroATX Mini Tower Case | $29.99 @ Mwave |
| Power Supply | EVGA 400W ATX Power Supply | $27.99 @ SuperBiiz |
| Wireless Network Adapter | TP-Link TL-WN781ND 802.11b/g/n PCI-Express x1 Wi-Fi Adapter | $14.50 @ Mwave |
| Prices include shipping, taxes, rebates, and discounts | ||
| Total | $241.43 | |
| *Lowest price parts chosen from parametric criteria | ||
| Generated by PCPartPicker 2015-04-06 12:00 EDT-0400 |
That said, if you can salvage an old case from somewhere (old PC, goodwill, old business PC's) you can shave off another $30
I also went for a 32GB SSD in lieu of a 320GB HDD. If you need space you can go for the HDD instead (its about $7 cheaper too)
Also, instead of Windows, you can use Linux. Windows is nearly $100, which would raise the price on this build by quite a bit.