I want to build a speedometer for a car using Arduino UNO. For the display, I'm using an LCD screen that will be located behind the steering wheel. For context, I'm building the speedometer for a Tesla, which does not have a speedometer behind the steering wheel. Because this speedometer is for a car, and for driving purposes, I need the speedometer to be very accurate. I am planning to calculate the current speed of the car using a GPS module, but not sure which GPS module would have the best accuracy but is not too expensive? I am also open to other methods that may be available to measure current speed.
I'm not sure what your price criterion is but a few months back I bought a u-blox neo6 on a BoB for I think about £13 UK though I can't recall the supplier. May be a shortage at the moment as the price seems a lot higher. Anyway I can recommend u-blox devices, used as OEM by may automotive companies.
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What accurazy do the regulations demand? Here, Sweden, Europe, the rules say: -0, +7% are the limits.
Know that GPS doesn't work under ground, in tunnels. Dence forest might also effect GPS devices.
can be within around 5kmh of the actual speed reported by the car
I'd say the coppers will soon let him know....... 
Op needs to read the regs on use of Arduino in certain circumstances...I'd say "out-of-bounds" straight up.
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So the car is allowed to +5 kmh? Hardly...
Only allowed +3 kmh here. 
I've found both commercial and my home built, NEO6-M based GPS, being really accurate.
Once the speedo in the bus got jamed and showed 75 kmh. Looking at the GPS, the speed was 120 kmh in the early darkness....
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