|Your specs include PCI Express expansion slot but has intergrated graphics:|
Ignore the below information as it is just food for thought. The more I looked at your set up th emore I realized that the HDMI graphics card may possibly be the better way to go for the money. I see they range from $46 to about $400.
Following link shows someone elses recommendations for your PC:
You do not need to purchase a Graphics card if your TV has a PC (VGA) Connection.
Amazon connectors from PC to TV:
You need a connector with an audio connector too.
Freezing and jittering would likely be attributed to the fact that you have a small amount of RAM and/or a poor internet connection to the laptop. (Likely the lack of RAM)
So unless the video card you are looking for has at least 1Gb of RAM then you will not do any better.
Crucial.com, for example, has a 2Gb Memory kit for your Dell Dimension E510 for $33 USD or even adding 512 Mb for $16.99 will give you enough to work with. (Crucial says "Comaptible" but I have tried to add "Compatible" Crucial RAM to my Dell and it was not so Compatible. It developed memory errors. Better going with a matched set and the next price jump in your favor would be the 2Gb for the noted amount.)
At home we use a laptop VGA to TV VGA with the audio input to my 5.1 home theater and it works very good to view anything displayed on the laptop's monitor.
For $99 you could get Apple TV which connects via HDMI:
or Western Digital TV for the same price:
I use WD Live TV but have since bought Apple TV as a gift for a friend as I suspect it is better.
(I spent too much time doing this)