What is the BER for multilevel modulated signal
This is not BPSK! BPSK is binary phase shift keying with binary meaning 2 levels. What the OP is describing is Pulse Amplitude Modulation where multiple bits are mapped to multiple levels. In this case there are 4 levels so it would be 4-PAM. (As Dilip points out in the comments that the levels in 4-PAM would typically be equally spaced while here the inner levels are spaced further than the outer levels, complicating the BER computation slightly as detailed below but still PAM regardless).

The demodulation scheme would similar to BPSK demodulation in that you would multiply the received signal with the recovered carrier and low pass filter. Decision would be made on the output by setting decision thresholds half way between each of the levels (assuming equiprobable data) and then the derivation for BER follows similar to BPSK (solving for the tail probabilities under Gaussian distributions if the noise is assumed to be AWGN as is typically assumed, noting the different tail contributions involved since the levels aren't equally distributed as Dilip has pointed out - that said the thresholds at ±1.5 will likely dominate setting the error rate with significantly smaller error contribution from the threshold at 0). 



With equiprobable distribution of symbols and AWGN with standard deviation $\sigma$ this would result in the following Symbol Error Rate:

$$\text{SER} = \text{erfc}(0.5/(\sqrt{2}\sigma) + \frac{1}{2}\text{erfc}(1/(\sqrt{2}\sigma) $$

For smaller $\sigma$ values the second term would likely be insignificant. For larger $\sigma$ there would be additional tail contributions from the next adjacent threshold levels which would also significantly complicate the BER computation as detailed below. Thankfully under all practical applications these tails would certainly be insignificant and can be safely ignored.

To go from Symbol Error Rate to Bit Error Rate, consider how many bit errors occur when a symbol error occurs. Because the mapping done by the OP was properly Gray-coded (only one bit change between adjacent symbols), only one bit error would occur with each symbol error.