In principle, a computer can compute $\tau$ for any knot, using [grid diagrams][1] (or [variants][2] [thereof][3]); Baldwin and Gillam [computed][4] $\tau$ for knots up to 11 crossings a while ago. However, this is not practical. Unlike with Khovanov homology, there are no programs that can compute even $\widehat{HFK}$ for "large" knots (say, more than 20 crossings).

There are a couple of tricks you can use, that sometimes help:

- Plamenevskaya gave a [lower bound][5] on $\tau$ using the self-linking number of any representative of $K$; more precisely, if $T$ is a transverse representative of $K$, then
$$ {\rm sl}\,T \le 2\tau(K) - 1.$$
Of course you can use it for both $K$ and its inverse, and see what you get. To generate some random Legendrian/transverse representatives of $K$ you can use [Gridlink][6].

- Another lower bound is given in terms of the unknotting number (I believe this is due to Ozsváth and Szabó):
$$ u(K) \le \tau(K),$$
and in fact there are more refined versions of this with signed unknotting numbers (for which I'm afraid I have no references, but I can try and dig them up).

- Of course, you can always try and simplify your knot using concordances, since $\tau$ is a concordance invariant (and then combine it with the previous tricks).

- Upper bounds depend a lot on the kind of information you have. Plamenevskaya's inequality also happens to give an upper bound (since it gives a lower bound on $\tau$ and $-\tau$); the Seifert and slice genus are obvious candidates, but the latter is usually quite hard to get your hands on.

[1]: http://msp.org/gt/2007/11-4/p09.xhtml
[2]: http://www.worldscientific.com/doi/abs/10.1142/S0218216510007796
[3]: http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=droz_j%2A&s5=floer&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq
[4]: http://www.worldscientific.com/doi/abs/10.1142/S0218216512500757
[5]: http://msp.org/agt/2004/4-1/p20.xhtml
[6]: http://homepages.math.uic.edu/~culler/gridlink/