Getting a doctor involved can help you time everything, so you're not wasting your time. It can also help you identify hormonal issues or other problems that you might not be aware of with a DIY insemination. I'm sure it happens...because there are stories like this out there:
http://www.nytimes.com/2013/06/02/fashion/My-Husbands-New-Son-Modern-Love.html?pagewanted=all&_r=0 "By the grace of the fertility gods, this one donation hit its mark."
But I've never actually heard of it being successful among anyone I know.
Much of the diagnostic testing will be covered by insurance, whether you have infertility coverage or not. I've heard from one woman who is a single mother that she had trouble with her insurance company because they wouldn't cover infertility treatments since she couldn't prove she had been trying for six months without success. Which obviously was impossible for a single woman trying to have a baby on her own, but whatever. Maybe trying DIY insemination for six months with your own donor would count?
A simple unmedicated IUI really isn't that expensive. This is from the CFA cost sheet last year:
$500 for a natural cycle IUI (without ultrasound or bloodwork monitoring.) That includes semen processing.
$1,500 for a Clomid IUI. That includes semen processing, the ultrasound and bloodwork after starting medication, and a pregnancy test. You're on your own for the baseline ultrasound and bloodwork -- but again, your insurance might cover that as regular old diagnostic testing.