How much does it cost in computing time/hardware to mine bitcoins?
It really depends on how much you're willing to spend.
Low end
Let's talk about the very smallest investment you could make for mining bitcoins: a Raspberry Pi. It can do 200,000 hashes per second, which might sound like a lot, but actually it's about 0.000001% of the total network hashrate.
Power: 3.75 W
Hashrate: 0.2 MHash/s
Cost: $35
(Note: this doesn't mean a RPi is useless for mining. You can use it as a host for your ASIC/FPGA miners.)
Mid range
Okay, CPU mining is bad, but what if we use GPU's?
Let's say we use a $162 graphics card, the 5830 from AMD. Let's throw in another $187 for case/motherboard/power supply. (Based off of this list I just threw together.)
Power: 175 W
Hashrate: 302 Mhash/s
Cost: $349
High end
Okay, let's get something specifically designed for bitcoin mining - the ModMiner Quad. It costs $1070, but it needs a host, so let's use a raspberry pi for that.
Power: 43 W
Hashrate: 800 Mhash/s
Cost: $1105
Ultra-high end
Okay, let's say that you have stupid amounts of money to use on mining hardware. Let's say that you can get a Bitforce Mini-Rig. (They are not being sold yet.) In fact, let's say that you can get 10 of them. This gives you 40% of the bitcoin hashrate.
Power: 15,000 W
Hashrate: 15,000,000 Mhash/s
Cost: $299,000
Sources for most of this
The lesson here is that you can spend any amount of money on bitcoin mining.