Is there a good online opening training software? There are a lot of websites to train your tactics, but I can't find any for opening repertoire. I am looking for something similar to 'Chess Position Trainer' but online, where I can import a pgn file with my repertoire, indicate that I want to train it as Black and track my progress. Does such a software exist?

UPDATE/NOTE: being able to train against my own reperoire, and rather than the best tree/stats move is vital for me.
2018 update:



It seems lichess.org offers a lot of the features you're looking for. On lichess, go to tools then choose Analysis board:


The editor allows you to import FEN, PGN or edit the board yourself. 
You have access to two proper opening databases, one extracted from the games played on lichess and one from the masters games (top players). It's not only an opening explorer showing the lines, but it also shows you samples of games where a given line is employed.
You can set up your position and play through the opening branches, while simultaneously verifying the engine assessment on the position. From a given position you can also choose to practice against the computer, for instance for a variation you've come up on your own.
Last but not least, once your analysis is done, you can save it all as a study (which can also be made private). As such, all the variations and side-lines you've explored/tried and plan to keep will be saved together with the engine analysis of them. These will be your personal repertoires, which you can continuously modify whenever you find improvements.


The features are available to everyone. Once you've created a couple of repertoires and started practicing them on lichess, in your profile there is an analytics page where all sorts of statistics will be calculated based on your performance. For example, your rate of win with a certain line, or your frequency of certain moves and so on. That will help you to assess how you are performing with your different openings.



Older suggestions:
In terms of simple online opening explorers, there's always the Shredder Opening Database, can be useful for quickly checking something.

Then there's the database available on chesstempo which is actually really detailed (statistics on each opening etc) and gives exemplary games of any opening played by real players. 

But I believe for what you want to do, lichess is the go-to choice here.