You would have a couple of different options, based on what you were sending and how urgently (and safely) you needed it to arrive. We're also going to assume that you are literate enough to write a letter, and that your family are literate enough to understand it when it arrives, which is not absolutely certain. If none of you are literate (or not literate enough to write a lengthy letter) it's possible you could find someone else to write it down or have it read for you.

Your first option would be to send someone you trust with your letter or things you need to send - maybe a friend, a merchant or a slave. This is probably someone who is going back to Italy for their own reasons and will be close enough to either pass it directly to your family or pass it to someone they trust who can pass it on. Obviously you'd also be sending them with instructions on how to find and recognise the person you're sending it to! We can see this in a letter from the Egyptian recruit Aurelius Polion who's stationed in Pannonia at some point in the third century. He has apparently given it to an unknown carrier to be handed to a veteran from his legion who will presumably deliver it on or have it collected from him. There are a lot of complaints in Roman letters about people failing to reply, but it's possible that a lot of these letters were simply lost along the way. Aurelius himself complains that he's written to his family a lot of times but they haven't written back and that he's hurt by them not responding.

Your second is to hire a courier to take your letter or parcel physically from one place to another, but this would be expensive; even the very fastest courier from Eburacum to Rome, (assuming good weather), would take almost a month. I'd recommend playing around with ORBIS to find out more about this. It would also have to be someone you trusted, especially if you were sending money - you would have no way of finding out if they had cheated you and taken the money you were sending home for a long time. You might be able to club together with some other soldiers to afford this, especially if you were all from the same area (which obviously would have varied over the period the army was in action - by the end of Roman Britain, it's not unlikely that your family would be ten minutes from the fort!).

The third is to use the cursus publicus, or Roman postal/courier system, which was created by Augustus. This, however, is not for private use, and is supposed to only carry letters about official business. As an ordinary Roman soldier stationed in Britain, you probably did not have access (especially for a letter to your family), although if you are an official with access you may have bent the rules to send your own notes. Typically one courier carried the message or parcel all the way (something which was much less efficient than the Persian system they adopted it from, which relied on relay riders). We know from various law codes that the cursus publicus did suffer from suborned couriers and permissions being sold illegally, so you may have been able to bribe a courier to take your message or parcel along with what they were supposed to be carrying.

Your fourth is to use the Roman military post, which is unfortunately not as well documented as the cursus publicus. It seems to have largely existed on the fringes of the Empire and to be mostly cavalryman couriers passing back and forth between camps. You might have been able to ask or bribe a courier to send a note to someone stationed in a nearby camp or pass a message on, but this wouldn't have been much use for sending a message all the way back to Italy, only to a next stop in a longer chain of transmission.