Meaning of "ask of"

This meaning of ask (definition 7) is mentioned in the OALD:

expect/demand
[transitive] to expect or demand something ...
ask something of somebody You're asking too much of him.

So, "doing everything you’re asking of it" means doing everything you expect or demand of it.

"doing everything you're asking of it" means "doing everything you're asking it to do".

Which, in the case of an inanimate thing, means "doing everything you're trying to get it to do".

'Asking of' here is in the same sense as 'demanding of.' Asking something to someone (or in this case, something - the browser) means putting a question. Asking something of someone is like making a request or asking a favour. It implies involvement of proactive effort on the part of the other party in the process of meeting your requirement(s).