I am not a professional colorist, but I had platinum hair for quite a few years, so I can give you a few tips!

Your hair looks pretty light already, like a level 8 or 9. If you keep bleaching, you will get close to the point of no return in regards to adding tone, there will be nothing for the color molecules to grab on to, so to speak. At this point, it looks like you only need to deposit, not lift.

First off, stay away from drugstore box kits, find a local beauty supply or someplace online to graduate to more professional supplies where you will have more control over developers, colors, etc.

If I were you, I would start to play around with toners, Demi-perminent dyes and temporary colors to add silver and violet tones to your hair to get a true platinum. The violet tones are opposite yellow on the color wheel, and will knock out a lot of the creamy vanilla tones and look more white/silvery.

A violet toned shampoo and conditioner, or adding a little bit of pure violet 'crazy color' dyes (semi-permanent, vegetable dyes like Manic Panic, etc.) to your conditioner are the way to go for low damage color. Manic Panic 'Virgin Snow' is a good start, it's a very pale, almost imperceptible violet toner, has no peroxide and will wash out in about 1-2 weeks.

Toners and Demi-perm dyes will use very low peroxide to add tone with little lift and damage.

Platinum is one of the toughest colors to maintain, which is why the violet shampoo/conditioner is key, I can't recommend them enough!

Your hair still looks healthy, but consider a once a week treatment of extra virgin coconut oil to counteract the bleaching. You can even coat your hair in it an hour before bleaching (without washing it out) for extra protection.

Hope this helped!! Good luck on your glamorous new platinum locks!