Everything I'm about to say is just my experience, so take it as you will! You will need two days of hair-hiding/not going out in public. I know you're just looking to cover your roots, but to lighten your roots, you'd have to bleach. So I'm just going to break EVERYTHING down and maybe it will be helpful to you or someone else lol.
So I went from dark brown (ombre) to blonde a few months ago. The brown was dyed, not my natural color (it's harder to bleach dyed hair than natural hair). Note, this is not the first time I've done so as I've dyed my hair regularly and multiple colors for 15 years. See the attachment for before/after photo.
So, for the initial bleaching, you don't want to use a box dye/bleach. Go to Sally's and buy these two things. If your hair is as dark as mine was, get a "40" on the liquid. Don't let the Sally's clerk talk you into less.
^ Get the Violet or Blue, not the White. If your hair turns brassy when bleaching, get Blue. If it turns yellow/orange, get Violet.
Ok, so those two things mix together to form your bleach. 1 scoop of powder for 2 scoops of liquid. Separate your hair into small sections when you apply and have someone help you, so you don't miss any areas. Start on the underneath of your hair, then the top length of your hair, then your roots on the back/underneath of your hair, then your roots at your scalp. When done, put a plastic bag over your hair to help build up heat, so it will set more quickly. Expect to leave the bleach on 30-45 minutes.
For whatever reason, regardless of how well you apply, there will be a few areas of your head that just do not take the bleach well and will not end up light enough even though the rest of your head is practically white. Go ahead and wash the bleach out and go to bed.
The next day (several hours later), apply more bleach to the spots that didn't take well the first time. Rinse out. Now your hair will be orange/yellow and generally look like crap. It's ready for your dye! If possible, wait several more hours before applying your actual hair dye color.
For the hair dye, I have tried the Sally's mix boxes as well as every cheap "box dye" brand there is. My favorite dye to use that has never disappointed me is Garnier Nutrisse. It's like $8 a box at Walmart/Walgreens/whatever. As a guide: If your hair is shoulder-blade length, get two boxes.
Here is a tip that you can take with a grain of salt. However, I find it to be the balance between crappy-looking home dye after bleaching and decent looking home dye after bleaching. Get a dye color that is the same as how your hair dyes. Like, I got a GOLDEN blonde because my hair bleaches with golden/yellow tones. If your hair bleaches more brassy, get a COOL dye, like ash blonde or something. Again, just a suggestion.
Other things you will need:
Purple shampoo -- This will "cool" (aka tone) the golden/orange tones in your hair each time you wash it. I recommend using it every other hairwash.
Protein packets -- Use this every other time you wash your hair for two weeks to help repair the damage. Don't skip these!
NO hairdryer/flat iron -- Painful, but your hair has been through enough. Keep it away from heat for a few weeks at least.
The reason you do the underneath length of your hair first with the bleach is because it will be sitting on your hair the longest because of the amount of time it takes sometime to put all the bleach on. Because of this, that part will have the most damage (so you hide it).
Important: The secret to not destroying your hair like you see some people with straw hair from bleaching is not to leave the bleach on too long. As in, this is why I tell you to go ahead and rinse it out if there are a few places that are still brown-ish. You can always apply again later. You don't want to just leave the bleach on for 2 hours to get those few spots lightened because it will destroy everywhere else that was already lightened.
Good luck!