You need to be reading a more beginner-oriented tutorial on bond maths.
Several of the details in your question are irrelevant: the mention of dv01 (although it might be the next step after you figure out the yield) and the fact that this note is the cheapest to deliver for some futures contact is also irrelevant. The fact that the note "last" traded at some price is only slightly relevant. You should be able to convert any price to yield and vice versa.
You ask what "the 5-1/8s of May 15, 2016" means. This means, the note having 5.125% annual coupon rate and May 15, 2016 maturity date. The combination of coupon rate and maturity is often used to identify a particular issue. I lookeed it up. FYI, it has cusip 912828FF2.
You need to know that the market convention for U.S. treasury notes and bonds is to pay semi-annual coupon equal to 1/2 of the quoted annual rate, i.e. 2.5625% every 6 months, and to repay the principal at maturity as a bullet payment. Other kinds of bonds have different market conventions.
The 19 cash flows of your note are:
11/15/2006 2.5625% coupon
05/15/2016 2.5625% coupon + 100% principal
Observe that we use "quasi-dates", not bumped for weekends or holidays. You need to know when to use them and when to use bumped dates.
Your remaining question is, given the (clean) price of the note, how do you calculate the yield. This cannot be done with the information you've provided because you did not include the settlement date. Your information is insufficient.
If you had the settlement date, then you would need to know how to discount the remaining cash flows using the given yield to arrive at the dirty price; how to calculate the coupon accrued from the start of the coupon perdiod to the settlement date using the note's daycount convention; and subtract the accrued from the dirty price to arrive at the clean price.
To go the other way, from clean price to yield, generally cannot be done using closed form. You need to try different values of yield until you match your price closely enough. Brent works well here. This is not practical to solve for yield manually, but you need to understand what the computer does for you.
There are also some special rules that apply only to price-yield of U.S. treasury notes and bonds (not to corporate bonds) during the last coupon period.
As noob2 said, you should walk through some good code in a debugger to understand how to project cash flows, how to calculate price from yield, and how to solve for yield given a price.