How to compute autocorrelation of signal defined by difference equations?
There are different approaches to compute the requested auto-correlations but I would like to provide the simplest that suits to your case provided that.


The difference equation is an LCCDE type indicating an LTI system.
The input to the system $v[n]$ is WSS (wide sense stationary) random process.


Both of which I assume to be observed in your question. Then I would use the following well known relation between the input and output auto-correlations of an (real) LTI system driven by (real) WSS input $v[n]$: 

$$ r_{dd}[k] = h[k] \star h[-k] \star r_{vv}[k] $$ 

where the auto-correlation of WSS $v[n]$ is $r_{vv}[k] = \sigma_v^2 \delta[k] = \delta[k]$ for an uncorrelated, zero mean, unit variance process. And $h[n]$ is the impulse-response of the LTI system signified by the LCCDE. 

Then the auto-correlation sequence of the WSS output $d[n]$ is
$$ r_{dd}[k] = h[k] \star h[-k] = \sum_{m=-\infty}^{\infty} h[m]h[-(k-m)]~~~,~~~\text{ for } k = 0,\pm 1, \pm 2,... $$

So assuming that your LCCDE :
$$d[n] - \alpha  ~d[n-1] = v[n] ~~~, ~~~\text{ with } |\alpha| &lt; 1 $$

signifies a causal LTI system, then its impulse response is:

$$ h[n] = (-\alpha)^n u[n] $$

and yields the auto-correlations to be:
$$ r_{dd}[k] = \sum_{m=-\infty}^{\infty} (-\alpha)^m u[m] (-\alpha)^{m-k} u[m-k]~~~,~~~\text{ for } k = 0,\pm 1, \pm 2,... $$

I hope you can proceed the rest.