You'd have to have a significantly-active blog to justify the costs of Azure. Aside from keeping a web role up and running (and just one instance chews up almost $90 monthly, as ZippyV stated), you'll also have to pay for data.
You do NOT need to invest in SQL Azure though. There's Azure table storage which is much better suited for your blog. It has a table structure, you can define entities (e.g. classes, maybe a BlogEntryClass) that are stored, and the storage costs will run you significantly less than SQL Azure (only $0.15 per GB per month, so your storage costs will likely remain well under a dollar a month for a blog, a small fraction of the cost of SQL Azure).
You'll also pay for bandwidth ($0.15 outbound per GB).
If this is a learning exercise, it's a great investment, but if you have an MSDN Premium account, you can host your blog there - you get 750 compute-hours monthly (enough to run a single role instance 24x7), 10GB table storage, and 14GB monthly outbound data.