How will multiple stages help, I think the next stage will just amplify the offset of the previous one and add some extra? Isn't offset mainly dependent on small differences in the semiconductor process and temperature?
I'd just go for an auto-zero op-amp like Dave did. A cheap way might be to try to trim (some of) the offset out, but you will have to adjust it if the temperature of the op-amp changes (which is basically a manual version of an auto-zero op-amp). Both Dave's video about the design of the ucurrent and his explanation of the auto-zero/chopper op-amp (this was an early one) would be helpful if you decide to go this route.
At 1uV, you will probably notice thermocouple effects and other fun stuff. Keithley has a 'Low level measurements handbook' that can be downloaded for free from their website. 1uV is on the upper side of what they consider 'low-level'. It contains lots of useful information, I have a hardcopy version that they send to me for some reason (not that I mind).
Another way would be to use a precision DMM, but this is definitely the more expensive route (between $25 and $250 or so). For example, the HP 3468A/3478A (about $50 to $150) has a resolution of 1uV at the 300mV range (but this is the limit, so if you actually want to measure 1uV with any accuracy, you need better). Keithley has a DMM in the 17x series that they call microvolt DMM, these can sometimes found cheaply on ebay, it also has a resolution of 1uV.