Yes, you can replace became increasingly important with became much more important and it will still be grammatically correct.
However, there is a lot going on in this sentence in terms of tense. We have the matter "growing in importance," personal computers "becoming more widespread," and operating systems "growing increasingly sophisticated." These all represent gradual processes, and my impression of the sentence is that you will go on to talk about a trend and not a specific moment in time. Making the change to became much more important, It suggests that you will go on to talk about a specific moment where the matter had already become much more important, but that you will talk about the other two issues (the widespreadness of personal computers and the sophistication of operating systems) as trends. If you want to discuss a specific moment in time, I would advise rewording all three to make it clear.
On an unrelated and unsolicited note, you should be aware that the spelling 'gonna' is only appropriate in the most informal contexts, like text messages to friends. Even if we say it out loud quite regularly, in writing is is generally to be avoided.