I am looking for a word that would mean somene who is always trying to climb higher on the social ladder. Kind of like a Hyacinth Bucket.
I guess social climber is one phrase, but is there another phrase. I was thinking “uppity woman” but that don’t really fit. Especially if it’s a man. 
Another word for Mrs Bucket might be “poseur”. Not sure if that’s what you’re thinking of though.
Arriviste?
That’s pretty close, but I don’t know if people would know what it means. Parvenu is kind of in that vein.
Is there more of a common word?
Now to me a “poseur” is someone who is faking it, instead of somene who is actually trying to aspire to it.
I don’t think someone like Mrs Bucket, was faking it, indeed poor Richard had to live with it.
bougie, or more correctly, petite bourgeois
Petite bourgeois are small business owners/shopkeepers.
In Spanish we have the word “trepa”.
It derives from the verb “trepar”, which means “to climb” (esp. to climb without external aids likes ropes and such – that is, to climb something using only your hands, feet and perhaps teeth 
).
A “trepa” is someone who will climb higher (at work, or in the social ladder), NO MATTER WHAT. A “trepa” will only think about him- or herself, will ruthlessly squash others under his or her feet to go up, and will use others left and right to achieve whatever manoeuvres allow for a speedier climb towards the top. And damn the consequences!
I don’t know if this would cover the kind of person the OT had in mind…