Depends on the company. Some companies look at your payment history with them only, some do a soft pull and some go all the way with a hard Inq. If you give us the name of the card you are trying to get a CLI on maybe someone who has the same card can tell you their experience.
Soft you can see when you pull your own report. It is not counted against you when calculating credit score.
Hard can be seen by anyone pulling your report (credit card company, bank, etc). Hard inquiries is a factor that affects credit scores(10% of your score depends on this). Too many hard inquiries in short period of time looks like you are desperate.
The thread below has a lot of information which you may find useful.
hsbc generally does a soft pull but you can call and ask. Also they are not very willing to give CLI, when I called they told me , "we do random reviews and when we see fit we'll increase your CL.