What is hasuji exactly? Hasuji is the path your sword takes in a cut and the edge alignment whilst in that cut.
'Good' hasuji happens when the path that your sword takes, points the edge of the blade directly forward and isn't tilted to one side or another, does not change angle throughout the cut and travels in a straight line from the start of the cut, to its finish. In formal japanese arts, hasuji will be described differently to this I am sure, but for the purpose of backyard or freestyle cutting, it is sufficient.
Now importance wise, Hasuji is probably the number one thing to train into your technique when you start cutting. If your hasuji is correct then the cut stands a much better chance of being successful. In fact, if your hasuji is spot on, then not only will your cut stand more chance of succeeding, it will take far less energy to effectively complete. As an added bonus, if you're lucky, you'll get a neat little whistle as the sword whips through the air.
In order to help improve hasuji it pays to work on your grip of the sword. Practicing your 'te no uchi' will certainly help but it is not limited to just this. Basically, the correct grip on your sword will guide your cut more accurately and more confidently, eliminating blade wobble and putting your sword where you intended it to go.
Large swings make sure that there is enough force in the cut, but making smaller swings with the sword means that there is less time for your cut to wander off track. It's also good form to not make overly large motions in a cut when smaller ones will do. This is something I'm still learning myself.
Leaning with the cut is sometimes practiced. Martial artists avoid leaning over with a cut as it telegraphs the intentions of the practitioner to his or her opponent. With competitive cutting however, this does not apply and some people have found that hasuji can directly be improved by aligning your body angle with the cut. Leaning over to one side whilst keeping the blade above the head for example allows you to see more clearly where your cut will go.
All these words however cannot improve your technique without practice and thats where it's at. Just carry on paying attention to the feedback your sword is giving you and you'll improve. A sword with bo-hi will give you much more feedback than a sword without and is especially useful for dry cutting but you can get a no-hi sword to whistle as well. it just takes that much more work and your technique has to be that much better.