I am using Bandicam for recording most of my gameplay, it's free and easy to use, lots of customizing options available as well. It allows for 10-minute videos at once, but if you press record right after the ten minutes are over, it'll just start recording again, so you can record whole matches with it. The only problem is that it puts a bandicam logo on top of the screen for advertising (it's not that big, only the website address, and it's quite understandable since you use it for free). If anyone can bring up better recorders, I would also be interested in them, though.
You can *ahem* "acquire" Fraps if you do some google searching, it's probably the best for pure recording. However just be aware that it records raw format and a video as sort as 5 minutes can take something ridiculous like 20GB.
Now, as for the OP's question, yes, if you're looking to both stream and record, OBS (what Xemko) linked will probably the best way to go about it unless you want to pay for Xsplit (OBS is probably better for smaller scale stuff though so you should be quite alright with it, it's what I use as well). There is... a plethora of youtube videos you can find about how to set up OBS so don't be afraid of it, though if you want some personal help, I can probably walk you through most of it in a matter of minutes as well.
Now, as for editing, that's where you're going to run into some problems if you don't have the budget to pay for a program. Stuff like Adobe Premiere and Final Cut Pro are pretty friggin expensive, Sony Vegas is probably the cheapest option you'd have for editing software, and even that's usually $60+ (unsure of the exact price, haven't looked it up recently).
Free video editing software you're pretty much going to be reduced to using either iMovie if you have a Mac, or Windows Movie Maker which, if you don't have I believe you can still download for free of Microsoft's website. Neither of those are going to offer you the levels of customization and well... pure options that paid for programs can of course, but they're... serviceable.
Good luck, let me know if you need any help!