I am writing a paper on dioxin, generally expressed as ppt, parts per
trillion. A research article I read lists the dose of dioxin given to
mice that increased the risk of death as 0.01 micrograms/kg.

Question: The toxic dose for a 70kg adult would be 0.7 micrograms/kg.
What would that dose be if expressed in ppt?  Thank you!

0.7 micrograms/kg  is .7 E-6g / E3g =  .7 E-9 to 1 = 700 E-12 to 1

so answer would be 700 ppt.

That is using 'scientific notation'  where e.g.  E3 = 10^3 = 1000    etc
http://www.about.ch/various/unit_conversion.html

It should be noted that according to dictionary
://www.google.com/search?hl=en&lr=lang_en&safe=off&q=trillion&btnG=Search
ttrillion is ambiguous and means

The cardinal number equal to 10^12 =  E12
Chiefly British. The cardinal number equal to 10^18 = E18

we are taking US meaning here. Fot this reason ppt is not SI unit and
should not be used:

 3. 	The combinations of letters "ppm," "ppb," and "ppt," and the
terms part per million, part per billion, and part per trillion, and
the like, are not used to express the values of quantities. The
following forms, for example, are used instead: 2.0 µL/L or 2.0 x 10-6
V, 4.3 nm/m or 4.3 x 10-9 l, 7 ps/s or 7 x 10-12 t, where V, l, and t
are, respectively, the quantity symbols for volume, length, and time.
(See Sec. 7.10.3.)
http://physics.nist.gov/Pubs/SP811/sec11.html

 In line with this microgram/kg is more propper unit then ppt.

1 ppt = 1 nanogram/Liter. You can't really convert this unless you
know the density of the liquid to convert the kg to liters. If you
assume it's around the density of water then the answer is 700 ppt.

ppt are units of concentration (eg, of a substance in water or air),
and micrograms per kg are units used to measure doses.
So in order to convert one to the other, you would also have to know
how much water was consumed.
If you are trying to measure the concentration of dioxin in living
tissue, then the answer above should be good.

There is considerable diversity and ambiguity in ways people measure
concentration; different fields and countries have different customs:

 Mass per unit volume, Percent by Mass (or mass fraction), Percent by Volume..
http://www.ilpi.com/msds/ref/concentration.html

 SI (international system of units) specifies what 'should' be used.
 and NIST, US institute which interprets SI units in US allows all fractions 
see: 8.6 Amount of substance, concentration, molality, and the like...

Table 12  summary description of nine quantities that are quotients
involving amount of substance, volume, or mass

http://physics.nist.gov/Pubs/SP811/sec08.html#tab12

I have used in my answer 
mass fraction
SI unit:   kg/kg = 1

based on context and advised asker to use mass fraction, which is
clearly defined, rather then ppt, which is ambiguous and used for
both,
 usually for mass per volume but sometimes also for mass fractions.