Relationship between SNR and link speed
Signal strength is only important in determining signal quality insofar as it affects the signal-to-noise ratio (SNR).  As the name implies, SNR is determined by dividing the signal power by the noise power in the same bandwidth ($\frac SN$).

Claude Shannon, the pioneer of information theory, came up with a formula that gives an upper limit to how much information (you can think of this as bits) can be transmitted for a given SNR and bandwidth $B$ (I will note, though, that Wikipedia calls it the Shannon-Hartley theorem).  The formula is as follows-

$$
C = B\log_2\left(1 + \frac{S}{N}\right)
$$
where $C$ is the channel capacity in bits/s, $B$ is the bandwidth, and $\frac{S}{N}$ is the SNR.  Modern transmitters/receivers try to get as close to this limit as they can through a number of means.  One way is to increase the bits per symbol as the SNR improves (e.g. going from QPSK to 16-QAM).  Another is to reduce the error correction coding as the SNR improves, and increase it when it worsens.