The natural test for a fair coin would be to compare the numbers of Heads and Tails, unless you wanted to include independent identical distributions in the null hypothesis as well (which you might if you did not conduct the test)
But you have said you want to look at the longest run
$6$ is rather short for the longest run in $1000$ tosses of a fair coin which is i.i.d. equally probably heads or tails. Since an i.i.d. unfair coin would tend to give longer runs, the doubt here could be that $6$ might come from a non-i.i.d process, possibly an attempt to fake a random sequence
You might intuitively guess that the longest run could be of the order of $\log_2(n)$ so about $10$ in this case ($10$ is indeed the median and mode here, and slightly below the expectation).
There are explicit expressions but it may be quicker to use a Markov calculation, producing something like the following, which suggests that at over $95\%$ confidence you might reject the null hypothesis when seeing longest runs of $7$ or fewer or of $16$ or more from a sequence of $1000$ results