If a postdoc is interested to apply for a future professorship in his/her current institution. What should she/he do in the postdoc research period? Based on the discussion in a previous question
Apply for assistant/associate professor at same institution as postdoc?
If someone is interested to apply for a future assistant/associate professorship in the same institution where one is currently working as a postdoc/research scientist. What one should do in his early stages of postdoc research to build independent scientific contribution so that the institution of interest would like to include him/her in its faculty members?
I could write a book about this, as I'm sure many faculty members could! My answer involves two major points: political reconnaissance and becoming intellectually self-assured.

First, I suggest you make some good acquaintances in the faculty beyond your PI and get an honest opinion on what it might take to be able to join it. The comments to your question about duplication are potentially valid, IMO, but the degree to which it matters or the precise interpretation is going to depend on the personalities involved, the political climate in your institution, and the tenure/promotion manual of the department/college/university. 

Of course, you just can't show up to the bar one evening and ask people point-blank after a couple of drinks :) You need to genuinely show interest in the direction and health of the department over an extended period of time, and possibly get more involved in departmental decision making, administration, or teaching. Diplomacy and politicking is key here.

Even if you don't get to formally ask some questions, thinking about them can help you detect subtle signs of their answers in the communications between faculty. E.g., Is there a prevailing mood that your PI's research area is popular among the faculty? Is it an area that the department even wants to expand? Do they even expect to be hiring anyone in the next 2-3 years? Is your PI going to be your champion and partner (you'll probably need that!) or is he/she going to perceive your presence as a colleague to be a source of departmental competition for lab space, students, money, attention, etc. I've known some PIs manipulate and exploit their post-docs so that they essentially become trapped in their position as essentially cheap technical labor (your experience with the PIs work is very valuable to them). There is usually a lot more going on privately that will affect the eventual decision to hire anyone new. Use the chance to reconnoiter and make alliances to your advantage while you are this close to the action. It may become clear that it will be much better for you to make a clean break, and you might get a stronger recommendation from your PI as a result.

Second, you have to plan early with your PI to show off your contributions above and beyond what the PI has done in establishing this particular line of research. If, between that and chats with other friends or colleagues, you can't figure out what you are or will be doing in the very near future that distinguishes your creative input and future personal direction as independent, then you won't be successful in communicating that in your writing or presentation to hirers.

IMO, your PhD study was supposed to be the time where you figured out a line of independent thought that you take forward during post-doc work. You should be dedicating time to develop those ideas into plans for projects over the next decade or more. You should use that to (diplomatically and respectfully) push your own agenda: you must stake out your space in your work and your perception in your community. Anything less and you are frankly setting yourself up for being labeled (fairly or unfairly) as not "faculty material". Gossip works the same in academia as it does in any walk of life, don't be fooled by the intellectualism.

If your PI is dominating your direction of work then get some side projects going of your own (even secretly), but discuss with your PI how to get more involved in the planning before it's too late! You won't last a minute in faculty-level academia without being assertive and developing healthy skills of persuasion. Fakers will be called out quickly, so get ready to defend your plans.

In any case, don't get your hopes up for staying on at the same institution. An academic career path is already extremely tough to succeed in without restricting your future options like this. I know first-hand how it is hard to imagine leaving a place you know and like but there are usually other good opportunities out there that you'll need to better appreciate. Stay brave and open-minded.