If they reproduce like normal humans, will their offspring be inbred, or genetically identical like everybody else? The only difference in any of the clones are the X and Y chromosomes.
If that original person was perfectly homozygous for all alleles--i.e., if both of every pair of chromosomes are absolutely identical--then every child will also be genetically identical to their same-gender parent, modulo de novo mutations.
If they are heterozygous, then sexual recombination will result in variations in the offspring--for any given gene, about half of the children in the second generation will be identical to a parent, but the other half will not. And given the large number of genes humans have with variant alleles, if the original is heterozygous for more than a couple of genes, it is essentially a statistical certainty that every child will be genetically different in some way, both from their parents and from all other children.
Whether or not it would cause problems would depend on exactly what set of recessive alleles might be carried in the original clone genome, but it needn't.
Logan above pretty well summed up the basics. For each dangerous recessive trait where the original individual was heterozygous, in the first generation of offspring, 25% of people would be homozygous for the negative trait. Some of these could be bad, but not totally crippling - like sickle cell anemia. It won't kill you (usually) but it sucks. 50% of people will be carriers, and 25% of people will be homozygous negative.
A lethal mutation (say, a null gene for an essential protein) will result in a 25% reduction in births (or a high infant mortality rate) with 66% of living people being heterozygous and 33% being homozygous negative.
If you have the tech for successful human cloning, you would (at least) be able to test the genotype of the people being cloned. With numbers like this, flaws will be readily apparent in the first generation of offspring. Those lucky enough to be homozygous negative for the bad traits will be sought-after as parents for the next generation - all their children will be healthy because they will always pass on one good gene. Heterozygous parents breeding with each other will face the same odds as the original clones - 25% sick, 25% dead, or both (depending on inheritance).
As long as there are not a large number of serious genetic flaws in the person, careful breeding for a few generations will get you past the initial crisis. Even heterozygous parents can make lots of embryos and screen for homozygous negative offspring. People may complain about this sort of screening from an ethical standpoint, but your clones won't really have a choice.
You didn't specify if you're talking about an initial population which then goes on to breed normally, or if every child is a new clone.
Unfortunately, in either case, Entropy is a Harsh Mistress. In the first case, random mutations will eventually doom your population due to accumulated errors. In the second case, you will need to absolutely perfectly preserve the original genome with no degradation, or the same thing will happen (perhaps more slowly). Even then, science fiction is full of examples where the second approach eventually falls apart.
You're basically dealing with the ultimate worst-case scenario of inbreeding.
The children can be different in all cases. A chromosome exists in 2 parts. If one has a ressessive defect (like colourblind), it won't show. But if 2 are present, ir simply one in males, they would be courblind (it's on the x chromosome and missing on the y, making any on the x automatically the only one for this defect)
Even if bith sides of the chromosome are identical and you get identical children, there can be differences. Even without random defects in the genes. Gene expression is important. Example: Plant corn in a field. Corn in the middle is well protected by othe other corn around it, allowing it to grow tall with small roots, as it's unlikely to be blown away. Corn on the edge is generally lower, as it's buffeted by the wind. It'll grow lower with deeper roots. Other genes have expressed themselves. So even in identical gene pool with identical chromosomes on both sides you can have differences.
As an icing on the cake, the genes that have expressed themselves are more likely to return in the next generation. A low growing corn will produce a lower corn in the next generation, even in the middle field. A human example is food. If food is scarce for a generation or a few people, the kuds will have more tendency to get fat if food is abundant than kids from identical parents that had enough food. The kids took the trait to store more energy in case of another hunger, making them fatter when they have plenty of food.
This last one is often not as dramatic as the others, but certainly present. This can make people in different cultures completely different without random mutation and just gene expression, although it is far more limited without random mutation and mixed gene pools.
The children wouldn't be identical, but they'd be very similar to each other. The founder presumably had a fairly typical amount of heterozygosity, and every locus he was heterozygous at, his clones' offspring would be 25% homozygous for one allele, 25% homozygous for the other allele, and 50% heterozygous. And since this is determined separately for different alleles, there'd be quite a bit of variation.
The closest real-life equivalent I can think of are inbred strains of laboratory animals, such as rats and mice. They're not produced by cloning, but by several generations of mating siblings together or line-breeding offspring with one parent (eg father/daughter, then father/daughter again, and so forth). They tend to be very similar, which makes them handy for research studies because it's more likely that any differences are caused by your experimental variable rather than just random variation. In fact, rats can smell pheromones to guess if another rat is related to them, and studies have indicated that members of the same inbred strain all smell like siblings to them.