Stellaris

You need to send out construction ships and build outposts (starbases) which costs alloys and influence to build. Influence is probably going to be your most limiting factor in doing this quickly. Build out incrementally, do not go out more than one system past your border. The influence cost is based on how far away from your border you are, one system out is 75 influence, two systems out from your border is 150, etc increasing 75 for each system further away.

To gain influence faster you need to have fleet size larger than your empire size. If your fleet/empire ratio is less than 1 your influence gain is scaled by that ratio. Declaring other empires rivals and using an Imperial Authority will increase the gain as well.

One other item of note, the smallest galaxy size will have exactly 200 stars so you need to clear out all the random alien dangers to get this.