It just happens to be the opposite for me, it becomes murch harder for me to portrait close family members.
But when I'm with "strangers" what I do is not making such a great deal with the shooting. I say, "this is just a test" (the first shots actually are) but then I never say "Ok, this is a real one", I simply make a natural transition from adjustment confirmation to real shooting without anouncing it.
In the time beetween shots, I make some non-prefabricated small talk, to connect with the client without getting too personal, being empathetic and non judgemental.
I also take little pauses while faking to stop paying attention to the portraitee. I pretend to review pictures in the camera, or check something in the equipment, but in relity, I pay close attention to their natural poses and gestures. After that, I tell them: "you took this posture" or "you laughed this way" and proceed to reproduce that, but this time sooting, making sure to tell them that the pose is coming from them.
It works pretty well on most subjects, because I'm not forcing them into postures or gestures they don't ussually make. I fisrt shot "as is" assesing the shot only at a glance in the camera's lcd. Then I improve the pose by making just small adjustments.
In the middle of that, take some unanounced shots when the client makes this "poses of their own". Do it quietly and don´t let them see you asessing the shot right after you took it, review it later.
In order to boost their comfidence, I show them a few shots quickly, only the ones I think are really good and only for a few seconds. Also I try my best to keep a "this is really good" face and not showing dissapointment if a shot is not perfect or something goes sligtly wrong. (A few days ago, for example, during a shot, one of my strobes misfired, I didn't make a deal out of it, and continued shooting. When I cheked in post, It just happens that the shot was perfectly good, albeit in a diferent mood bacause of the lighting. I simply processed it accordingly).
To put it briefly, I play a bit of a psychoanalyst, and try my best for the client to feel comfortable and comfident, so their personality comes to surface. It works almost with every one, with some quicker than others. Sometimes allowing the presence of a family member or close friend or co-worker may help if that person happens to be a confident booster for the protraitee, but if they to the opposite, take them out of the room ASAP.
Your own attitude plays a big role too. Being comfident and positive yourself relieves part of the scenic stress for the client. Do not let it show if you are in a rush, or if you are concerned with something else.