By RAW an alchemist don't have a caster level, so he doesn't qualify for the requirements to take any crafting feats.
You can read more about it in this FAQ which states:
Is an alchemist a spellcaster for the purpose of crafting magic items other than potions?
As written, no, alchemists are not spellcasters, and therefore can't select feats such as Craft Wondrous Item.
The design team is aware that this creates some thematic problems with the idea of an alchemist creating golems and so on, and plan to examine this in the future.
That said, if you really want to craft magic items you could try to:
- Obtain a spell-like ability with a caster level high enought to qualify for the feat (usually the racial spell like have a caster level equal to your caracter level, so those are handy) or
- Take the master craftsman feat at level 5, and then the craft wonderous item (useless, since your wizard buddy already has it, probably 4 level sooner) or craft magic arms and armors
The feat Create Portal was deleted by the Player's Guide to Faerûn, but prerequisites for creating portals remain otherwise unchanged
The Player's Guide to Faerûn on Changes to Feats from the Campaign Setting says
In addition to overhauling the regional feat system, this chapter also introduces a small number of changes to other feats from the Forgotten Realms Campaign Setting. Most of these alterations are minor changes to the wording for clarity, but some of the key changes include the following.…
Create Portal: Portals are now considered wondrous items and are built with the Craft Wondrous Item feat. The new Portal Master feat makes it easier for a character to build a portal. (42)
(The Player's Guide likewise describes the feat Portal Master (42).)
So by removing the feat Create Portal, any ol' caster with access (somehow) to the correct spells and a feat with a prerequisite of caster level 3rd can opt to create a portal.
Thus, without portals requiring their own specific feat to create and the large number of high-level casters on Toril and the utility value of the feat Craft Wondrous Item, the ubiquity of Faerûnian portals becomes a bit more understandable, despite portals themselves still possessing sort-of silly yet otherwise untouched prerequisites for creation.
So you know, Underdark has a few more portal qualities (52-3), Expedition to Undermountain describes Halaster’s drifting portals (218), and Power of Faerûn and Dungeon #92 describe a portal that emits deadly flying needles (152) and coral portals (79), but these latter two designs are largely specific to the location and group associated with them, respectively.
Note: This revision means that the DM must make minor adjustments to the prerequisites of the item creation feat Create Moving Portal and the epic feat Create Time Portal.
Best Answer
Potentially any spell can be tattooed
First, for your specific case of having Burning Hands tattooed on your hand, note the existing Spell Tattoo which works in a similar way to a scroll and is just a one time use. The price listed for this tattoo is:
So for a normal Burning Hands of caster level 1, the market price would be 25x4=100 gp but creating it would cost half this price (50 gp) since Inscribe Magical Tattoo follows the rules for magic item creation.
If you don't want to have this burning hands spell to be single use you can also use the option of creating a new tattoo:
By consulting the section on pricing magic items, you can figure out how much it would cost to create a tattoo of a spell with more ways to be activated (remembering that as tattoos count as slotless items they would always cost double).
Since no limitation is placed on which spells can be inscribed this way, any spell can potentially be tattooed with DM approval.