[RPG] create a computer game based on the Pathfinder core rules

OGL: Yes. For details, see the faq:

Q: I want to distribute computer software using the OGL. Is that possible?

A: Yes, it's certainly possible. The most significant thing that will impact your effort is that you have to give all the recipients the right to extract and use any Open Game Content you've included in your application, and you have to clearly identify what part of the software is Open Game Content.

One way is to design your application so that all the Open Game Content resides in files that are human-readable (that is, in a format that can be opened and understood by a reasonable person). Another is to have all the data used by the program viewable somehow while the program runs.

Distributing the source code not an acceptable method of compliance. First off, most programming languages are not easy to understand if the user hasnÍt studied the language. Second, the source code is a separate entity from the executable file. The user must have access to the actual Open Content used.

See the Software FAQ for more information.

GSL: No. It only lets you do certain things, and software's not one of them.

3. Licensed Products. The license granted in Section 4 is for use solely in connection with Licensee’s publication, distribution, and sale of roleplaying games and roleplaying game supplements that contain the Licensed Materials and are published in a hardcover or soft-cover printed book format or in a single-download electronic book format (such as .pdf), and accessory products to the foregoing roleplaying games and roleplaying game supplements that are not otherwise listed as excluded in Section 5.5 (“Licensed Products”).

5.5 Licensed Products. This License applies solely to Licensed Products as defined in Section 3 and to the specified uses set forth in Section 4. For the avoidance of doubt, and by way of example only, no Licensed Product will (a) include web sites, interactive products, miniatures, or character creators; (b) describe a process for creating a character or applying the effects of experience to a character; (c) use the terms “Core Rules” or “Core Rulebook” or variations thereof on its cover or title, in self-reference or in advertising or marketing thereof; (d) refer to any artwork, imagery or other depiction contained in a Core Rulebook; (e) reprint any material contained in a Core Rulebook except as explicitly provided in Section 4; or (f) be incorporated into another product that is itself not a Licensed Product (such as, by way of example only, a magazine or book compilation).

Also see the FAQ where they tell you "No" some more.

Solo Itself

I don't think there are any house rules required to play Pathfinder one-on-one. There are solo adventures for Pathfinder from Expeditious Retreat Press and have been for other D&D versions over time as well. All that's required is DMing. Since there's one PC, they will need difficulties tuned to them and some kind of safety net in case of bad luck - a NPC friend, or fate points, or something.

Investigation

If you are concentrating on more storytelling and whatnot and it's low combat then most of the problem isn't getting killed so much as getting stumped. If you're going to do lots of investigation consider Lorefinder, the Pathfinder/GUMSHOE mashup, which changes investigation skills to not block PCs. I playtested it and it brings the good parts of GUMSHOE (Robin Laws' investigative system that powers several Pelgrane Press games) to Pathfinder.

Advancement

In our Pathfinder games we stopped using XP a long time ago; the DM advances us when it's time. That removes the grind incentive and allows story development without artificial combat.

FATE

We also about half the time use FATE aspects and points in Pathfinder. Essentially we declare three aspects to our character, and when one gets us in trouble we get a point, you can spend a point for a reroll or a +4 if it can be linked to an aspect. FPs reset to 3 at each level. I am ambivalent about how much they really help with roleplaying, but they do present a consumable resource PCs can use when they think it's important.

Relationships

In terms of NPC relationships, I have a separate question open to get good relationship mechanics for Pathfinder but no one's come through yet really.