Hiring a vehicle to go there will be very expensive as only high-axle 4-wheel-drives such as Land Cruisers can handle the roads and they must be equipped with extra fuel tanks and the full kit of survival gear. You can of course join a tour, but northern Kenya tends to be popular with wealthier tourists and the package prices reflect that. Also, due to the relative unpopularity of the area, you may need to wait a while for a group to be assembled, and you will be committed to the tour operator's itinerary.
Fortunately, Lake Turkana is accessible by public means on both sides. If you aim for the eastern shore of the lake, your objective is the town of Loyangalani. To get there, you should catch a matatu from Nairobi to Nyahururu, and transfer there to a matatu bound for Maralal. This leg can be done in one day if you start early.
Since there are no buses going north from Maralal, you will need to hitch a ride on one of the big trucks that carry freight up to the lake. These trucks routinely carry passengers so there is nothing unusual about doing this. The ride is bumpy and you are exposed to the elements, but the views are incredible. To find one of these trucks in Maralal town, ask some of the men hanging around the main roundabout or the petrol station to put you in touch with a driver. These men will actually come to you when they see you arrive in town. You should pay them a finder's fee of 100 or 200 shillings if they successfully find you a ride. The actual fare is set by the driver of the truck, but the guy who finds you the ride can help you to negotiate a reasonable price.
It may seem like a long process to get there, but you have to weigh that against the time you could waste while waiting for a seat to become available on a group tour. That is assuming there are tour groups going at all.
The western shore of the lake takes less time to reach because there are direct buses and flights to the town of Lodwar. From there you need to travel a couple of hours to reach the lakeshore. I haven't been this way myself but I believe hitching or catching a local "bus" (which could just be a pickup full of tribespeople) would be options. If you are booked into one of the lakeside lodges they may pick you up from Lodwar but there could be money involved.
Lake Turkana is absolutely beautiful and it's worth the trip!