I recently attended a lecture given by the CEO of a private company.
This company has a number of large contracts, has received VC
investment from large, reputable firms (Motorola Ventures, The Tokarz
Group, etc). I believe the comapny would be a good investment.
Is it possible for private individual to invest a modest amount
(~$10,000) in a private company?
If so, what is the appropriate way to go about doing so?
Thank you for your time.

It is possible, but generally given the circumstances it won't be
permitted by existing obligations that were assumed at the time the VC
made their investment.

Generally speaking, the VC will now sit on, if not control the board
of director's of the organization upi are looking to invest in.  There
is no way to invest in a private company without the consent of the
board of director (this is the foundation of the difference between
"public" and "private").  The place to start is with the company's CEO
and the question "are you currently seeking additional funding".  If
the CEO has the leg room to make such decisions after their initial
rounds of funding then it may be OK.  But...do be warned that if a VC
is OK with you getting in on the action it may be a sign that the
believe the deal has soured - it means they are not willing to risk
any more of their own money.  If they think its a good deal a VC WANTS
to invest more in later rounds, or to get another VC in on the action
and build alliances (all very generally speaking).

That said, 10,000 is usually not worth the effort required (legal,
beurocratic) for a company to get some investment, unless very early
rounds.

Alternatively, you could approach the VC and see if they are still
seeking investment in the fund that is buying into the company you are
interested in.  While that will dillute your investment across all
companies the fund is interested in, it might be a nice option.  That
said...most VCs do not accept money in sums as small as 10K.

The short answer is "unless you have some connection to the firm or a
TON of money or unless you are taking on an invesment that has gone
very high risk then it is unlikely that investment attempts will
succeed.

I agree with all the above, just to point out that it is not just one
person's opinion.
Reading Maqrchbald-ga's comment reminded me of another point.  A
private company interested in additional capital of $ 10,000 would
probably be able to raise it from existing owners, a very much simpler
matter for the company.

I can understand your interest; some private companies can be very
attractive, but if they are, they have no need for an additional new
owner.
Sorry.  That is what the stock market is for  - for us little investors.
Myoarin