What is the significance of certain black organizations spelling
Africa with a 'k' (Afrika)?  I think this might be the German
spelling, but why would American organization use such a spelling?  Is
there an implicit political statement in this spelling (such as when
people spell America with a 'k')?

I believe you are quite correct in speculating that there is an
implicit political statement (of kinship and solidarity with the
continent in question) in the spelling 'Afrika' when it is used by an
American group.

Here you will find a discussion of this subject, attributed to writer
Haki Madhubuti:

"We are not certain of the origin of the name Afrika, but we are sure
the name spelled with 'C' came into use when Afrikans were dispersed
over the world. There the 'K' symbolizes our coming back together
again."

Trinicenter: Why Spell Afrika with a K? 
http://www.trinicenter.com/kwame/2002/Feb/172002.htm

This essay specifically addresses reasons why an activist organization
might choose the spelling 'Afrika':

"In the spelling Afrika, a 'k' is used rather than a 'c' because for
many activists the 'k' represents an acknowledgment that 'Africa' is
not the true name of that vast continent. When one speaks of Afrika,
they're bringing an Afrikan-centered view to the meaning. Therefore,
the Afrika spelled with a 'k' represents a redefined and potentially
different Afrika, and also it
symbolizes a coming back together of Afrikan people worldwide. Let it
be understood that when one speaks of Afrika and when most whites
think of 'Africa,' they are coming from two different worldviews. One
view supports the Afrikan ethos, while the other view supports the
European ethos."

South Jersey Grad: The Struggles of Afrikans in America
http://www.geocities.com/southjerseygrad/afrikanstruggles.txt

Here the Pan-Afrikan Center of Augsburg College explains its choice of
the spelling 'Afrika':

"PAN-AFRIKANISM is the movement to achieve solidarity and unification
of Afrikan people who have been historically separated and oppressed
by imperialism, racism, colonialism, and the slave trade. It uses an
intellectual approach to address the social, political, and economic
conditions that face Afrikan people throughout the Diaspora.

The DIASPORA refers to the areas (including the continent) where
Afrikan people have been dispersed, primarily by European colonial
force. Spelling AFRIKA with a 'k' is a symbol of self-definition."

Augsburg College: Pan-Afrikan Center
http://www.augsburg.edu/pan-afrikan/mission.html

Musician Darrell Rose discusses why he chose the spelling 'Afrika':

"Afrika Phase Three lasted for two years. We played a variety of
traditional Nigerian rhythms and idioms, which we also tried to fuse
into a new idiom--a new understanding of the past as perceived by the
present. That was the whole idea behind the name and the music. Africa
was--and is--in its third stage. The colonization and exploitation of
the last two centuries separates modern Africa from its proud and
ancient past. It hasn't erased it--although that was the intent of its
exploiters--but it sharpened divisions and created artificial
barriers. That's why we spelled Africa with a 'k'--it was an effort to
reclaim that which had been taken from us."

Camera Obscura: Healing Rhythms
http://www.hicom.net/~oedipus/darrell.html

Here's an unusual and politicized explanation:

"Why We spell Afrika with a 'k': Before de arrival of european
kkkolonialist, Afrika was originally spelled with a 'k'. De continent
gets its name from a Brotha who was emperor of de ancient Zingh Empire
over 15,000 years ago. Emperor Tirus Afrik is also responsible for
placing de red, blak, and green flag all over de continent, this is
how de continent came to be named after him. It wasn't until after de
arrival of de Italian general Scipio Africanus who conquered Afrika
for de Italians many centuries later, that de hard 'k' was replaced
with de letter 'c'. So Our spelling Afrika with a 'k' represents Our
conscious awareness of Ourselves and Our social development as a
Peepoe, and Our conscious desire to once again come together in unity
as one Peepoe, one Nation, with one destiny; with one flag: Red, blak,
and green!"

Lene Pantawapirom's Homepage
http://home4.inet.tele.dk/lepan/lene/indiana/jun00.htm

My Google search strategy:

Google Web Search: "spelling" + "afrika" 
://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=spelling+afrika

Thanks for asking an interesting question. I hope the information I've
provided is useful. If anything is unclear or incomplete, or if a link
does not function, please request clarification before rating my
answer, and I'll gladly offer further assistance.

Best regards,
pinkfreud

Beautiful answer.  This will really help my research.  Thanks so much.

I always thought that it was spelled Afrika because that's the correct
original spelling in the native dialect. For instance the National
Anthem of South Africa is "Nkosi Sikelel’ iAfrika" and there are
versions sung in a variety of Bantu languages like Xhosa, Swahili and
Zulu which all spell Afrika that way. I guess my idea of it was just
simplistic and logical. I never thought it had anything to do with
politics. I just thought the anthem was beautiful in all of those
languages and I love the choral music from Africa as well.