Can a UK and US dual citizen travel on a US Passport Card?

If you are a person with dual nationality and with two passports, one form each country for which you are a citizen, then you can travel using either passport (you only need to use one) - however, if you enter a foreign country using one passport you must use the same passport when leaving the foreign country.

You would use your UK passport to enter British territory and your US passport to enter US territory

For other teritories, you would select and use the passport which gives you the least visa requirements - and you would be able to call on the services of both the UK and US embasies wherever you were.