Can we use the present continuous form to make a general statement?

The continuous forms are used in English to convey quite a lot of subtextual implications.

The LearnEnglishLanguageWell site gives some relevant examples:

• John’s always losing his keys. (annoyance, shown by exasperated tone)
• Mrs Smith is wonderful. She's always giving people vegetables and flowers from her garden. (admiration)
• I’m forever seeing Mary on the bus these days. (could be annoyance or neutral, unmarked)
• My dog‘s continually escaping from the garden. (probably annoyance)

Also, the continuous construction is used to talk about things which are unexpected or unplanned. Examples (again from LearnEnglishLanguageWell) are:

• I always meet my husband at the station on Fridays after work. (a regular, planned arrangement ... no surprises: present simple used)

• We‘re always meeting Bill and Florence in the supermarket. (unplanned: present continuous)

The same happens in the past tense (/ past continuous):

• Our English teacher always organised monthly trips to the theatre. (planned, regular: past simple)

• Our parents were always organising surprise picnics and trips to the coast. (unexpected, irregular but iterative: past continuous)

However, in your example "I'm always trying to improve my English" - the action does not seem to be unexpected, so according to this site, it would not be recommended to use it in this way.

In the Cambridge Dictionary one finds a different focus:

We can use *always* with continuous verb forms
to refer to regular events or states, 
especially ones which are problematic or which we do not like or want

However, the examples it gives only reference annoyance.

In a British Council site I found these examples

[S]omething which happens again and again: Note that we normally use always with this ['iterative'] use:

• It's always raining in London.

• They are always arguing.

• George is great. He's always laughing.

My conclusion:

In all these examples, the continuous aspect conveys an extra nuance, compared to the simple aspect, which is more neutral. Therefore, if you want to emphasise your persistence, I think it is definitely okay to use I'm always trying to improve my English. If you just want to state it as a neutral fact, then I always try to improve my English is enough.