nightcraw1er.488: I tend not to keep anything personal on the computer, but on removable media.
This remember for a few years back, i was in visit at an company, while i was leaving, in the parking lot, there was an USB flash stick lying around. Took it, tried it out in an PC, it was having all kind of work-related stuff on it and, around all that, an folder called "DO NOT OPEN" or something like it. Of course i opened it, just to find out some nude pictures of a girl with who i talked earlier. I did not returned that stick.
Now, to answer the OP, clear browser data and move your personal data over. Now, depends what do you understand by "personal data". I keep all kind of stuff on my personal PC, from my bank accounts to passwords, scans from my personal documents, so, of course, i would remove those, put them on an USB flash, if i would let anyone else handle my PC. But other than that, i wouldn't care for other things and, trust me, the guys that handle your PC don't usually care either, like photos or stuff like that.
Other than that, most companies that do this kind of work need to have some kind of confidentiality agreement. Your battery problem might need the HDD to be in, with an working operating system. They might need to check the levels, how fast does it drain, it's easier from OS. If you bought the laptop from them, most likely you are not allowed to open it up and remove the HDD and, let's be frank, it's not as easy to do as on an desktop PC. If you don't know what you are doing, you might mess things up. So, you better don't worry too much about it. The technicians don't usually have time to search over your computer and watch your holiday pictures. They pretty much do their job and that's it.