The short answer is, the rules suggest ignoring these kinds of “fitting” problems, and therefore do not provide much in the way of rules for solving them. They do touch on some niche cases that might prove useful as examples of how your DM might houserule a modification process, which I’ve detailed below. But ultimately both Pathfinder’s authors and myself personally recommend against such houserules:
When an article of magic clothing or jewelry is discovered, most of the time size shouldn't be an issue. Many magic garments are made to be easily adjustable, or they adjust themselves magically to the wearer. Size should not keep characters of various kinds from using magic items.
Moreover, the Alchemist itself doesn’t suggest this should be a problem:
The tentacle is fully under his control and cannot be concealed except with magic or bulky clothing.
Magic clothing interacts with the tentacle in special ways just because it is magic. Seems to me that magic armors, much as they resize so the stockiest dwarf can wear the armor of the twiggiest elf, also can accomodate the tentacle.
So really, this is not supposed to be an issue, the rules don’t explain how to deal with making it an issue, and it seems an unnecessary nerf to the Alchemist class. If your DM is houseruling this, he also has to houserule how such modifications may be made and what they cost. The rules don’t cover it.
Full Plate
The rules only acknowledge “fitting” issues in the case of Full Plate; in all other cases one-size-fits-all-in-Size (that is, anything that fits a Medium humanoid-ish creature will fit any other Medium humanoid-ish creature). For Full Plate,
Each suit of full plate must be individually fitted to its owner by a master armorsmith, although a captured suit can be resized to fit a new owner at a cost of 200 to 800 (2d4 × 100) gold pieces.
No DC is given, though “master” might imply “someone capable of making masterwork items” which is a DC 18. Alternatively, it would be odd if such modifications are more difficult than making the armor from scratch, so just use the original DC. The cost is exorbitant for anything but Full Plate, since Full Plate is far-and-away the most of expensive of the core armors. On the other hand, the values range from just over ⅛ to just over ½ the base cost of the armor, so maybe those might be useful guidelines.
Armor for Unusual Creatures
There are rules for nonhumanoid characters’ armor, but in this case they’re talking about things like horses, which need a great deal more armor in a completely different style, not just an extra slot for an arm. At any rate, here are those rules:
Armor and shields for unusually big creatures, unusually little creatures, and non-humanoid creatures (such as horses) have different costs and weights from those given on Table: Armor and Shields. Refer to the appropriate line on Table: Armor for Unusual Creatures and apply the multipliers to cost and weight for the armor type in question.
There are no rules for converting regular armor to unusual-creature armor. Considering the creatures that these rules are intended for, such modification is probably impossible.
Best Answer
Mechanics (Intellect)
The exceptions called out for mechanics are cybernetics (which may need a joint Medicine check) and anything that needs extensive programming (which needs a Computers check). Explosives aren't either of these.
Inventing new types of explosives would be a Knowledge check, as would understanding a novel type of explosive, but manufacturing explosives through a known process is Mechanics. Making use of general-purpose explosives on the dramatic scale -- that is, when you must be precise in time or effect -- is also Mechanics. Setting an explosive trap you don't want other people to find is more a matter for Skullduggery.
I wouldn't recommend demanding two Mechanics checks to craft an explosive for purpose and also use it for that purpose, outside of a scenario where you wind up with a strongly split result on the first roll. Fail with a bunch of advantages and you'll need to actually do it live, redoing the roll with boost dice = 1/2 advantages. Succeed with a bunch of threat and you'll need to fine-tune it live with a (hopefully lower!) difficulty = 1/2 threat.