the previous answer is correct.
You need to check the properties of the video dropped in the timeline. (right click on the vid, select "properties") then you need to check the project properties in the file menu. (file>properties)
Now when rendering the video, you want it to be as close as possible to the original size & quality. If the original is a tiny size like say 640x360 & you try to render it larger, that stretching it can make the quality go bad. I'm not sure where you got the video in the first place, as far as if it's an mp4, wmv, avi, etc. or the size, but you can use a screen recorder while playing the video portion you wish to drop in the timeline to render.
Resize the playback & recording to at least 1280x720 or larger. Whatever screen recording software, be sure the settings record hq as well. the original video needs to be a decent quality & size to begin with. If you use a screen recorder in hq, then if it records in say an AVI that won't drop in SV, simply first use a free program like freemake to convert the clip to a friendly format say mp4, then use this converted clip to drop inside SV. There are lots of free screen recorders but usually the free versions have a limit on how long you can record, which might be fine for what you're trying to do. After dropping it to SV after converting if needed, then go to the pan/crop in SV & right click in the middle of the window to "match output aspect" so the size doesn't have black borders you won't want in your final render. (if you do this, be sure you're in the starting point of the vid first - otherwise it will resize as the vid plays through)
steps using a screen recorder
1. Record screen of vid clip you want.
2. convert to mp4 using www.freemake.com use the same (original size) option.
3. drop the newly converted clip of the mp4 inside SV
4. in the timeline click at the beginning/start of the vid (or clip off the ends that you don't want by using the "s" key to make any slices 1st.)
5. go to the right end of the vid click on the square icon pic this brings you to the event pan/crop window.
6. Again, make sure you're all the way over to the left at the start of the vid
7. now right click the middle of the window, select "match output aspect" so all of the video is inside the screen with no black borders.
8. if you need to center/align the window, do so now.
9. double click on the video in the timeline, to be sure the little yellow markers up above have one all the way on the left where the vid begins & the other all the way over on the right where the vid ends.
10. hit "render as" & be sure you have "render loop region" selected, then locate the MainConcept AVC/AAC (*mp4) & scroll down to at least the Internet HD 1280x720 & name the file where you want it saved & the render should be perfect & clear. If you have a SV version that allows you to customize, click in there after hitting "render as' & be sure in the video tab you choose "best" for the quality.
this should come out perfectly clear.
Also I believe when you upload a vid to YT, it downgrades the quality some, so I'd choose the minimum size/qual of 1280x720, if your clip is very short & you have high speed internet (so it doesn't take forever to upload to YT) you can opt to render in 1920x1080 but remember, your initial recording would also need to be that same size or larger to begin with.
In your question you seem to indicate the problem to be with your computer but it's more about a problem with the original video quality & size.
I hope this helps! If you finally do successfully post up the vid you wanted, you might take a few moments to add a note to this question so we can see how great it turned out!
I hope you have great results :0)