Green Dot secured (and every other secured card) is going to expect to be treated like a credit card. They will have a minimum payment each month, they will charge you interest (on your own money! I know! 
) if you carry a balance, and so forth. It's not like a refillable debit card, though it sounds like you're going to be wise and treat it like that—spend, pay to the card, spend, pay to the card—lather, rinse, repeat.
I had a Green Dot Platinum secured and I always paid it in full each month and never paid any fees (other than the annual fee) or interest. I paid via debit card online and the payment usually hit the card (thus freeing up some of the credit limit) in a day or two. For the platinum secured, I used this link to pay: https://platinum.greendot.com/ I would never, never, ever try to pay via checking account because I have heard they are glacially slow to post the payment. Debit card payment is the way to go. (They used to even let me pay with my Discover IT credit card, but no more, I don't think! 
)
Edit: You mention paying at convenience stores: I think you can do that, but I only did it once (to give my initial $200 secured deposit). I paid in cash to Walmart and paid a fee. You can avoid paying fees in the future if you pay online with your debit card.
Hope this helps! I really liked Green Dot and they helped me with my credit rebuild.
