Kasrik, Urban Bounty Hunter

The background I picked is Urban Bounty Hunter from the Sword Coast Adventure's Guide. It's not so much of a background as a current occupation, as Kasrik is the kind of private investigator people hire to 'find' (and often dispense with) criminals, scum and other villains who are difficult to pin under the weight of the law.

The background is official, so there's not much to discuss, but I do have one question (and also think you should know about the main benefit it grants). First the question:

It says I begin with a set of clothes appropriate to my duties, as those duties include ruffing up ruffians, can I trade in the Padded Armor I normally get as a level 0 commoner for a Chain Shirt (which, as a Dwarf, I am proficient at wearing)?

Secondly, the main ability (that you should just be aware of, as I'm certain to lean on it from time to time):

Ear to the Ground: You are in frequent contact with people in the segment of society that your chosen quarries move through. These people might be associated with the criminal underworld, the rough-and-tumble folk of the streets, or members of high society. This connection comes in the form of a contact in any city you visit, a person who provides information about the people and places of the local area.

What is the difference between armour classes and cost, I couldn't really find a chain shirt unless it's only in SCAG which I don't have

Chain shirt is in the PHB and in the free 'try this game' documents. Helpful image below!

Padded costs 5gp, Chain Shirt costs 50 gp. I'm also forgoing the opportunity to take any kind of exceptional clothing (such as an explorers outfit, valued at 10 gp). The net mechanical benefit is a +2 AC. I'd also be happy to 'weaken' it and make it 'Low Quality Chain Shirt' and tack the Disadvantage trait that some other medium and heavy armour has. I could also toss in the 5gp remaining from my background case and the free simple weapon for being level 0 (since I'm using my background cash to buy a warhammer).

So, since you asked about gold cost, I broke down all the financial details of my build so far (as you can guess by now, I do enjoy mathing out mechanical details).

I get Padded Armor (5gp), 1 simple weapon or tool (max value of a weapon is 5 gp, as it seems 'cheesey' to say I'm trading in artisan's tools or something which I believe are worth much more), a 'suitable' outfit from my background (looking at 'reasonable clothing' that would be 10 gp), and 20 gp cash. The net 'gold value' here is 40gp

I am looking to have a Warhammer (15gp) and Chain Shirt (50gp), both which Dwarves are proficient with (so it is stuff the character could conceivably use). That's 65 gp, so I guess I am asking if the 'professional fit' criteria on my background granted gear can explain 25 gp. That, having done the numbers, seems like a bit much.

A Battleaxe is 5gp cheaper (and the only difference is it is slashing instead of bludgeoning damage), and the difference then would only be 20 gp. Yet another option is 2 light hammers (4 gp total, so 6 gp even cheaper), making the difference only 14 gp.

Disadvantage appears to be roughly valued at between 5 and 350 gp, looking at the table (it's the difference between Scale and Breastplate, 350gp; and also padded and leather; 5 gp), so adding that on would reduce the value of the chain shirt, but how much is unclear.

So, the three options I proposed then are (in order of preference):

1. Warhammer + Low Quality Chain Shirt (25 gold difference, less 'low quality')
2. 2 Light Hammers + Low Quality Chain Shirt (14 gold difference, less 'low quality')
3. Battleaxe + Low Quality Chain Shirt (20 gold difference, less 'low quality')

It's also worth saying that I won't benefit from the Dex bouns to AC chain shirts allow. I'd prefer Ring Mail (heavy armor) which is 20gp cheaper than a chain shirt, but unfortunately I don't get racial proficiency so can't wear it. Chain shirt is the next best thing (as clearly half plate is too expensive, at 750 gp ;) ).

I should of looked in the phb instead of a quick Google search but the chain shirt plus the war hammer is 65 gold, the overall you have is 30 odd, I'm guessing you're Dex will be over 2 so if we cut down the shirt to a plus 1 and then have 10 or so gold to upgrade it to the +2 version should be fine, don't want to put too many disadvantages on something

Was writing a reply while looking up the phb and stuff

My dex is actually 11, so you can make the max dex bonus 0 and I wouldn't mind ;)

Screw it, you're not actually benefiting from the Dex bonus so yeah it's fine