Anyone can write fanfic.  However, there's a big difference between writing fanfic and writing good fanfic.  If you put your fic on the Net, you want it to be read, right?  So here's some advice on how to do it.

1)  Know your stuff.  Don't just randomly throw in some character names and situations, make sure these are the people and places that readers will recognise.

2) Dialogue.  Do the characters sound like they do on screen?  Re-read your work and really imagine each character reciting the dialogue.  If it's not them then change it.   For example, Data saying, "Ya think it'd work?"  Or Danziger: "I have a great amount of faith in your ability to generate a solution to the current dilemma."  Um...right, Danz.

3) Names.  Make sure the characters call each other the right thing.  Sure, you might have your private pet names for the characters you adore, but would Janeway really call him "Chuckles"?  I think not.  Jack calls Sam "Carter", she calls him "Sir" or "the colonel".  And while the characters might have nicknames for each other, don't overuse those either.  I think Jim called Blair "my little guppy" exactly once onscreen yet it shows up in fanfic all the time.  In Stargate, "Danny" is only used in extreme situations and rarely at that.  Get the names right!

2)  Terminology.  Make sure you know whether it's Medlab or Sickbay or the infirmary or the med tent or simply a hospital.  Nothing throws a reader more than the wrong word.

5) Language and spelling.  Pick either British English or American English and stick with it.

6) On the other hand, make sure you use the correct english terminology for the characters and situations.  20th century Americans use miles.  The rest of the world (and future time frames) use kilometres.
 

So...you might be wondering...how do I figure out what the right terminology is?  How can I possibly remember what the entire crew of Voyager calls each other?  There are two very basic ways to do this*.

1)  Re-watch as many episodes as you can.  Live the show.  Watch one per day or in one hit until you feel you know the characters and their mannerisms backwards.  Take notes if you have to, but I'll tell you this now: a lot of the information you seek will sink into your mind unconsciously.  (It happens to me all the time, I don't "know" the term for something but when I write, it's there.)

Of course, watching episodes takes a lot of time which many of us don't have.  There is a way to get around this.

2)  Read fan fiction by other people.  But do it with a critical eye - you are not reading for enjoyment, you are reading to learn.  At the same time, if a fic particularly touches you, then you should tell the author.  Feedback is always appreciated.

You should also note that a lot of fanfic will recount a character's history which saves you an awful lot of work in research.  These histories have been painstakingly pieced together by many authors until they become like urban legends: everyone knows that Blair and Naomi travelled a lot, that Jack was suicidal before the Stargate program, that Devon came from a wealthy family and was a "princess", that Mulder's sister was abducted....

When reading fanfic in a fandom, take note of the styles used.  See what's been done, and what hasn't been done.  And once your research is complete, throw it out the window.

Yes, you heard me.  Forget everything you read and write from your heart.  Write in your own unique style.  It can be poetry, or wild sci-fi, from something that would be seen onscreen to something completely out there.  As long as you have the characters (and terminology, to a lesser extent) correct then readers should love what you're writing.  Please don't recycle the same plot, or style, that you've seen over and over.  Even if you are writing something that's been done a thousand times before (like Mulder and Scully finally admitting they love each other) make sure you've put your stamp on it.  Make it uniquely different and yours.

Trust me, the readers will love you for it.  There are far too many "formula" fics out there.  Don't fall into the trap.
 

* (A third way to do research is to visit fandom-specific web sites, but it's often difficult to find the exact information you're searching for.  How many fan sites do thorough character dossiers rather than technical/historical information?)