However, I could only can understand 30% of what they sayid.

この場合、過去形を使った方がいいです。You used "watched" in the previous sentence, so you should also write this sentence using past tense.

With English subtitles, I think I can understand about 70% of them[what they said].

Today, I watched the PBS news hour, which[for an] hour, [because] I think that it[PBS] covers more serious topics than other TV programs do.

Be careful here. "which" should modify PBS, but you used it to modify "hour." Use "because" to explain a reason, and "which" to describe.

There, I encountered the[a] expression that I don't know.

"There's no proof Trump was seeking to help himself in the 2020 election." In this sentence, does "help himself" mean "do something for his own personal aim"?

I know [the usage ofat] "help oneself" [can be used] in the following way: "Please help yourself." We use this in food and drink context.

Today, II often (sometimes) watched the news in order to build up my vocabulary and improve my listening comprehension.

This is a statement of a habit or routine so you should present tense. This also makes your tense in the first, second, and third sentences consistent.

However, I can only can understand 30% of what theyis sayid.

As another user wrote, be careful when using words like they, he, it etc. My correction shows one way to avoid this issue.

Today, I watched the PBS nNews hHour, which I think that it covers more serious topics than other TV programs (do).

However, I can only can understand 30% of what they say.

Since we understand that by "they" you mean the anchors and other people who come out on the news, it isn't necessarily wrong to use it would a referent. However, you should try to make sure that you have clear referents for your pronouns when possible! You could use "However, I can only understand 30% of what the anchors say" or just make it passive and say, "However, I can only understand 30% of what is said."

With English subtitles, I think I can understand about 70% of them.

Today, I watched the PBS news hour, which I think that it covers more serious topics than other TV programs (do).

I put your final "do" in parentheses because it isn't necessary but it sounds fine to native speakers whether you do or don't include it.

There, I encountered thean expression that I don't know.

"There's no proof Trump was seeking to help himself in the 2020 election." In this sentence, does "help himself" mean "do something for his own personal aim"?

I know the usage of "help oneself" in the following way:phrase, "Please help yourself." We use this in the context of having food and drink contexts.