I play such a different game from most groups. First, I have created a GMPC, not because I wanted to, but because I was initially going to be a player then our GM basically quit and asked me to do it. The group said they would like my character to remain in the group. I do observe a few rules to minimize the intrusive possibilities of the GMPC. Basically, I identify any ways I might be inclined to show favoritism to my character and then do the opposite. So i frequently fudge die rolls to ensure that the GMPC fails saves, takes damage, etc.. This does a couple of neat things, first, it keeps players from suspecting favoritism. Second, it ups the ante a bit by making dangers a little more real when the group might otherwise escape unscathed.
Another thing I did was design the DMPC to be a support character. He's a Bard. So party buffs, skill-heavy, utility spells, and esoteric skills are what he provides to the group. He is essentially incapable of stealing the show! Further, he's not very flashy. He fires arrows in combat. That's all he does and all he's good at. Other characters get options, heal, cast, sneak attack, smite, channel energy...but not the GMPC. His job is to be vanilla. In fact, I frequently have him remain silent during RP time too. He pops in with jokes from time to time, but he's rarely instrumental in building alliances or solving problems unless the PC's will really benefit from a knowledge(weird shit)check.
This has happened before too. I've had the group form a strong attachment to one of my NPC's and have them join the party. I'm proud I can create characters that they love, so I often end up with DMPC's even in games where I didn't have one to start. It's never been a problem so far, but I think that's because for me, it really is all about making sure the PC's have fun and that's all I use the GMPC's for.