I understand that these fields overlap quite a bit and I am interested in both. How difficult would it be to double major in them at a typical university.
Each uni will have requirements for each major and what’s allowable for dbl majors. Your asking of this broad question presumes some universal standard. It doesn’t exist. You’ll have to investigate school by school
How is it at your school?
You may simply take related courses that you are interested in, but I don’t think double major in these two highly overlapping field would gain you anything.
I want to do it mostly for personal interest. So I guess, I could live without the double major. 
If your school allows you to do it and it doesn’t take you any extra time, I don’t think that there’s anything bad about double majoring. I do not think that it’d be highly beneficial to have two degrees from these fields but it certainly won’t hurt you!
Well, can you do it Rice U.?
Google says yes! You also could major in one and minor in the other.
There is so much overlap between cell biology and biochemistry that they’re virtually the same major.
Um… have you looked at their website? It’s a BS/BA in Cell Biology AND Biochemistry.
The reason they do this is most undergrads are not going to delve into the real topics too heavily. It’s easier with biochem simply because of the chemistry basis. But a general undergrad is not going to do a courseload heavy in signaling, virology, trafficking, and genetics. (I know this because I did it. Most people at my level at schools even better than mine haven’t done it.) There is a limit to what a normal undergrad can understand. Only someone who is uncharacteristically dedicated will attempt to take that high level science courseload.
Just to let you know, the name means very little. Their BS in Cell Biology and Biochemistry is exactly the same as Iowa State’s BS in biochemistry. And I do mean EXACTLY the same. If you want to talk to someone who has a degree in biochemistry with minors in microbiology and genetics, an REU, 2 years of research at ISU, and an Amgen Scholar position at one of the northeast locations (Harvard/MIT), PM me. I can talk about the 6 graduate courses I’ve already taken, my research, goals, and perspective.
I did look at their website, but I was unsure whether that meant two separate degrees or just one degree. But from what you are telling me (and considering the overlap) it means only one degree
I’m not sure I see the point in this double major. The two majors are similar enough that you can enter a grad program in either or better yet go to medical or other professional school. Otherwise they both have lousy career prospects either on their own or combined. A useful double major would be something like business or computer science with one of the two so that at least you have a plan B when like most who get science degrees, they desperately seek careers in unrelated fields due to the previously mentioned exceeding poor career prospects.