Let me point out that this post is concerned with making a music mashup video, which involves is two or more songs mashed together like this. This post is not concerned with mashup videos like this, where the mashup is of two or more films or television programs that don't accompany a song.
Approach 1 - Use the official videos of the songs in the mashup.
In mashup music videos like this the video has been created by combining scenes from the original videos of the songs used in the mashup. To create a video like this you will need a video editing software application, your mashup audio file and digital file versions of any videos you wish to use in your mashup video. Once you have these things you can set about making your video.